Reena Raggi, Circuit Judge:
Defendants Joseph Rutigliano, Peter J. Ajemian, Peter J. Lesniewski, and Marie Baran stand convicted in the United States District Court for the Southern District of New York (Victor Marrero, Judge ) for participating in a massive scheme to defraud the United States Railroad Retirement Board ("RRB") by filing fraudulent disability pension applications on behalf of Long Island Rail Road ("LIRR") employees. See United States v. Ajemian , No. 11 Cr. 1091 (S.D.N.Y.), Dkt. Nos. 401, 592, 649, 704. Defendants here appeal from a subsequent judgment of the same court, entered on November 1, 2016, which, in response to their motions for vacatur of sentence pursuant to 28 U.S.C. § 2255, amended each defendant's sentence in part to reduce original restitution obligations: for Rutigliano from $82,356,348 to $42,317,076; for Ajemian from $116,500,000 *101to $53,494,797; for Lesniewski from $70,632,900 to $34,237,476; and for Baran from $31,398,907 to $21,467,954. See United States v. Ajemian , No. 11 Cr. 1091, 2016 WL 6820730, at *3-4 (S.D.N.Y. Nov. 1, 2016). Defendants argue that even these reduced amounts are not supported by record evidence. The government cross-appeals, arguing that § 2255 does not authorize, nor do the facts warrant, any reduction of the restitution parts of defendants' sentences. The government further argues that the certificate of appealability granted defendants in this case does not reach their restitution challenges; that Ajemian waived any challenge to restitution in his plea agreement; and that, insofar as Baran had already filed one unsuccessful § 2255 motion, she had not secured the necessary leave to file a successive petition.
We need not reach the government's last three arguments because we are persuaded by its first. We conclude that § 2255 does not authorize district courts to hear collateral challenges to the restitution parts of criminal sentences where, as here, restitution obligations do not equate to custodial punishment. Accordingly, we vacate the § 2255 judgment reducing defendants' restitution obligations as imposed in their original judgments of conviction, and we remand for reinstatement of the original judgments. To the extent defendant Baran also invoked coram nobis to seek relief from restitution, the district court did not address that claim, but no remand is necessary for that purpose because Baran cannot show fundamental error in the original order of restitution.
BACKGROUND
I. Defendants' Convictions and Original Sentences
The fraud scheme for which defendants stand convicted sought to take advantage of a disability pension plan made available to LIRR employees by the RRB. The LIRR's own pension plan allowed employees with 20 or more years of service to retire as early as age 50 and to receive payments equal to half their pre-retirement income. When an employee reached age 62 or 65, he was eligible for an additional retirement pension from the RRB. The RRB also offered disability pensions to employees of any age who were no longer able to work, for which payments began immediately upon approval of a disability claim.
As part of a decades-long scheme to defraud the RRB, defendants Rutigliano, Ajemian, Lesniewski, and Baran, together with 17 confederates, repeatedly submitted applications for disability pensions based on fabricated medical documents. Orthopedic physicians Ajemian and Lesniewski created these fabricated documents. Rutigliano, a former LIRR conductor and union local president, obtained a disability pension for himself by submitting false documentation of his physical condition from Lesniewski. Rutigliano also acted as a facilitator or consultant for numerous other LIRR employees seeking disability pensions, charging approximately $1,000 per person to fill out fraudulent applications. Baran, a former RRB employee whose LIRR-employed husband obtained an RRB disability pension through a fraudulent application supported by Lesniewski, also acted as a facilitator-for-hire, submitting disability pension applications for LIRR employees that she knew were supported by fraudulent documentation.
On January 18, 2013, Ajemian entered into an agreement with the government and pleaded guilty to conspiracy to commit mail, wire, and health care fraud, as well as substantive health care fraud. See 18 U.S.C. §§ 1347, 1349. On May 24, 2013, the district court sentenced Ajemian inter alia *102to 96 months' imprisonment and ordered $116,500,000 restitution, the amount identified in his plea agreement. Further consistent with that agreement, Ajemian did not challenge his conviction or any part of his sentence on direct appeal.
Rutigliano, Lesniewski, and Baran stood trial and, on August 6, 2013, a jury found them guilty of multiple conspiratorial and substantive counts of mail, wire, and health care fraud. See id. §§ 371, 1341, 1343, 1347, 1349. The district court sentenced Rutigliano and Lesniewski inter alia to 96 months' imprisonment each and to $82,356,348 and $70,632,900 in restitution respectively. It sentenced Baran inter alia to 60 months' imprisonment and $31,398,907 restitution.
The restitution amounts ordered were based on data compiled by Natasha Marx, an Auditor in the Office of the RRB Inspector General, and reflected disability pension payments made to identified LIRR annuitants-for whom each defendant had submitted or supported fraudulent claims-as of the dates the RRB terminated their disability pensions due to the discovery of the fraud.
Rutigliano, Lesniewski, and Baran unsuccessfully challenged their convictions, including their sentences, on direct appeal. See United States v. Rutigliano , 790 F.3d 389 (2d Cir. 2015) ; United States v. Rutigliano , 614 Fed.Appx. 542 (2d Cir. 2015).
II. The RRB Board Orders and Post -Termination Approvals
Defendants submit that post-conviction actions by the RRB constitute new evidence supporting the instant collateral challenges to their convictions and sentences. We briefly summarize those actions here.
On June 27, 2013, i.e. , approximately one month after Ajemian's conviction, the RRB issued Board Order 13-33, which terminated the disability pension benefits of all LIRR employees whose applications were supported by medical evidence from Ajemian. The RRB provided notice to all annuitants affected by the order, and termination became effective three months after the date of those notices. On September 30, 2013, i.e. , approximately two months after the guilty verdicts against Rutigliano, Lesniewski, and Baran, the RRB issued Board Order 13-55, which similarly terminated the disability pension benefits of all LIRR employees whose applications were supported by medical evidence from Lesniewski. By their terms, Orders 13-33 and 13-55 terminated benefits only prospectively; they did not seek disgorgement of benefits paid prior to their pronouncement. Further, the orders permitted employees whose benefits were terminated to reapply based on "current medical evidence" of disability from within twelve months of the new application and not derived from Ajemian or Lesniewski. Lesniewski App'x at 168-69, 171-72.
By May 22, 2015, the RRB had decided 531 reapplications for disability pension benefits, granting 485 and reinstating such benefits retroactive to the terminations effected by Board Orders 13-33 and 13-55. Approximately 80% of the granted reapplications relied on at least one new medical condition not included in the employee's initial fraudulent application, and approximately 70% of the granted applications omitted at least one medical condition relied on in the initial fraudulent applications.
III. Collateral Challenges to Convictions and Sentences
A. Baran's First § 2255 Motion
On July 15, 2015, Baran moved pursuant to 28 U.S.C. § 2255 for vacatur of her conviction based on ineffective assistance *103of counsel. The motion made no challenge to any part of Baran's sentence. Specifically, it did not challenge restitution although the RRB had, by then, taken the aforementioned action on disability pension reapplications. The district court denied Baran's § 2255 motion on January 5, 2016. See Baran v. United States , 160 F.Supp.3d 591, 594 (S.D.N.Y. 2016). Baran did not appeal.
B. Defendants' Rule 33 and Instant § 2255 Motions
On October 2 and 13, 2015, Lesniewski and Rutigliano respectively moved for a new trial under Fed. R. Crim. P. 33, arguing that the RRB's approval of the vast majority of reapplications for disability pension benefits was newly discovered evidence showing that there was no fraud in the original applications. Alternatively, they moved for resentencing under 28 U.S.C. § 2255 on the ground that the RRB's approval of reapplications showed that the district court had based defendants' original sentences-both their incarceratory and non-incarceratory components-on erroneously high loss calculations. On November 25, 2015, Ajemian also cited the RRB's approval of reapplications to move for vacatur of his conviction and resentencing pursuant to § 2255.
On March 4, 2016, the district court denied Rutigliano's and Lesniewski's Rule 33 motions, see United States v. Rutigliano , 168 F.Supp.3d 661, 667 (S.D.N.Y. 2016), which denials this court subsequently affirmed, see United States v. Rutigliano, 694 Fed.Appx. 19, 22, 24 (2d Cir. 2017). On March 18, 2016, the district court denied Ajemian's § 2255 motion for vacatur of conviction, see Ajemian v. United States , 171 F.Supp.3d 206, 209, 215 (S.D.N.Y. 2016), and on June 21, 2016, it denied all defendants' § 2255 challenges to the incarceratory parts of their sentences, explaining that the alleged reduction in actual loss to the RRB was immaterial because defendants' Guidelines ranges were properly calculated by reference to intended loss, see United States v. Ajemian , 193 F.Supp.3d 298, 302-03 (S.D.N.Y. 2016), aff'd sub nom . United States v. Rutigliano , 694 Fed.Appx. 19 (2d Cir. 2017).1 Nevertheless, recognizing that restitution may only be ordered with respect to actual loss, the district court directed the parties to submit additional briefing on the question of whether the RRB's reapplication approvals in fact showed that it suffered a lower actual loss amount than had been thought at the original sentencing. See id. at 303-04.
Approximately two weeks earlier, on June 3, 2016, Baran had advanced essentially the same arguments as Rutigliano, Ajemian, and Lesniewski to move for resentencing pursuant to either § 2255 or a writ of error coram nobis under 28 U.S.C. § 1651. The district court denied the motion on June 29, 2016, to the extent Baran challenged her incarceratory sentence, but directed that she too file supplemental briefing on whether a reduction in restitution was warranted. See United States v. Baran, 192 F.Supp.3d 496, 501-03 (S.D.N.Y. 2016).
On November 1, 2016, the district court granted all four defendants' § 2255 motions in part, amending their original judgments of conviction to reduce their restitution obligations by the amounts of all pre-termination disability pension benefits paid *104to individuals whose benefits reapplications the RRB had thereafter approved. See United States v. Ajemian , 2016 WL 6820730, at *3-4. The district court reasoned that the reapplication approvals were based on current and reliable medical evidence that employees qualified for benefits without regard to earlier fraudulent submissions, and that the RRB was not entitled to restitution for past payments it was thus required to make in any event. See id. at *3. The district court explained that this reasoning did not apply to disability pension recipients who had not sought or obtained reapplication approvals and, thus, it rejected defendants' argument for vacatur of all restitution obligations. See id. at *4. Accordingly, the district court reduced defendants' restitution obligations as indicated earlier, so that Rutigliano was now obligated in the amount of $42,317,076; Ajemian in the amount of $53,494,797; Lesniewski in the amount of $34,237,476; and Baran in the amount of $21,467,954. See id.
The parties timely filed these cross appeals, challenging only the district court's ruling with respect to the reduction of the restitution parts of defendants' sentences.
DISCUSSION
I. Jurisdiction to Review Defendants' § 2255 Challenges to Restitution
We begin with the government's argument that defendants could not use § 2255 to challenge the restitution parts of their original sentences, because the issue implicates the district court's subject matter jurisdiction. See Kaminski v. United States , 339 F.3d 84, 86-87, 91 (2d Cir. 2003) (affirming dismissal of § 2255 challenge to restitution order for lack of subject matter jurisdiction). As this is a question of law, our review is de novo . See Moser v. Pollin , 294 F.3d 335, 339 (2d Cir. 2002).
Section 2255 states in pertinent part as follows:
A prisoner in custody under sentence of a court established by Act of Congress claiming the right to be released upon the ground that the sentence was imposed in violation of the Constitution or laws of the United States, ... or is otherwise subject to collateral attack, may move the court which imposed the sentence to vacate, set aside or correct the sentence.
28 U.S.C. § 2255(a). It is well established in our precedent that to secure § 2255 review, a petitioner "must satisfy the jurisdictional 'in custody' requirement" of the statute. Scanio v. United States , 37 F.3d 858, 860 (2d Cir. 1994) ; see United States v. Brilliant , 274 F.2d 618, 620 (2d Cir. 1960) (holding district court "without jurisdiction to entertain ... § 2255 motion if the relator or movant is not in custody"); see also Vega v. Schneiderman , 861 F.3d 72, 74 (2d Cir. 2017) (reaching same conclusion with respect to parallel "in custody" requirement of § 2254).
At the time defendants filed the § 2255 motions at issue on this appeal, all were "in custody" serving the incarceratory terms of their respective sentences. Thus, to the extent defendants' § 2255 motions challenged their prison terms, the district court had jurisdiction to consider those challenges on their merits. See Kaminski v. United States , 339 F.3d at 86. Whether the district court was also empowered to review defendants' challenges to restitution presents a different question.
In considering that issue, we are guided by this court's decision in Kaminski v. United States . We there concluded that the monetary components of criminal sentences, such as fines and restitution orders, generally do not restrict liberty so severely as to satisfy the custody requirement *105of § 2255. See id. at 86-87. Kaminski explained that § 2255, by its terms, applies to prisoners "claiming the right to be released"; thus, the statute "may not be used to bring collateral challenges addressed solely to noncustodial punishments" such as fines and restitution. Id. ; accord Gonzalez v. United States , 792 F.3d 232, 237 (2d Cir. 2015) (reiterating that § 2255 motion generally "may not attack the non-custodial aspects of a sentence").
A party cannot avoid this conclusion by challenging both the custodial and noncustodial parts of his sentence in the same § 2255 motion. In Kaminski , as here, the incarcerated defendant's § 2255 motion challenged both his prison term and his restitution obligation. This court held that the district court properly exercised jurisdiction over the prison-term challenge, see Kaminski v. United States , 339 F.3d at 86, but "lacked subject matter jurisdiction" over the restitution challenge because the defendant "was not in custody in relation to the claims against his order of restitution," id. at 91. In sum, a noncustodial punishment "may not be attacked in a § 2255 [motion], even if the [motion] also alleges error in the sentence of imprisonment." Id. at 88-89 (identifying "no reason why the presence of a plausible claim against a custodial punishment should make a noncustodial [restitution] punishment more amenable to collateral review than it otherwise might be"). Accordingly, to the extent the challenged restitution orders here are noncustodial punishments, precedent dictates that the district court lacked subject matter jurisdiction to review such challenges under § 2255 even though the same motions challenged defendants' prison sentences.
In nevertheless defending the district court's exercise of § 2255 jurisdiction, defendants argue that their restitution orders should be deemed "custodial" because the amounts are so high. The argument cannot be dismissed out of hand because this court, by contrast to most of our sister circuits, has not pronounced "fines and restitution orders to be, ipso facto , noncustodial" so as categorically to preclude their § 2255 review. Id. at 87.2 Rather, we have left open the question whether such fines and orders "could ever be such a restraint on the liberty of a petitioner as to amount to custody." Id. ; accord Gonzalez v. United States , 792 F.3d at 237. The question is grounded in precedent recognizing that even a non-incarcerated petitioner may satisfy the "in custody" requirement of the federal habeas statutes by showing that he is subject to "a significant restraint upon [his] physical liberty 'not shared by the public generally.' " Vega v. Schneiderman , 861 F.3d at 74 (quoting Jones v. Cunningham , 371 U.S. 236, 240, 83 S.Ct. 373, 9 L.Ed.2d 285 (1963) ). This does not mean that every aspect of a sentence not shared by the general public equates to custody for purposes of § 2254 or § 2255 review. Rather, the challenged condition must "impose a severe restraint on individual liberty or the imminent threat of such a restraint."
*106Id. ; see Kaminski v. United States , 339 F.3d at 86-87. Thus, the Supreme Court has recognized parole to constitute a sufficiently severe restraint on individual liberty to keep a defendant in the state's "custody" as that word is used in the federal habeas statutes. See Jones v. Cunningham, 371 U.S. at 243, 83 S.Ct. 373. But a sentence of conditional discharge may or may not so severely restrain liberty as to equate to custody depending on whether it requires a defendant's "physical presence at particular times and locations, both for community service and court appearances," Nowakowski v. New York , 835 F.3d 210, 217 (2d Cir. 2016) (holding defendant in custody in such circumstances), or only his refrainment from certain conduct, such as contact with a specified person, see Vega v. Schneiderman , 861 F.3d at 75 (holding that conditional discharge requiring defendant to abide by order of protection did not equate to custody under § 2254 ).
While, in Kaminski , we left open the possibility of a restitution order imposing such a severe restraint on individual liberty as to equate to custody, this court also observed that such situations will likely be "rare" and, in any event, "cannot be known" until the "terms" of restitution, as well as the "amount," are set. Gonzalez v. United States , 792 F.3d at 237. Thus, when in Kaminski we held a $21,180 restitution order not to equate to custodial punishment reviewable under § 2255, we focused on the terms of payment, which were there limited "on a monthly basis [to] the greater of ten percent of Kaminski's monthly income, or $100," and concluded that those terms did "not come close to" creating a sufficiently severe restraint on liberty to equate to custody. Kaminski v. United States , 339 F.3d at 87. Following Kaminski , district courts in this circuit have dismissed § 2255 challenges to even multi-million-dollar restitution orders based on similar payment terms. See, e.g. , Mansour v. United States , No. 11 Cr. 612, 2015 WL 1573327, at *4 (S.D.N.Y. Apr. 9, 2015) (holding challenge to $9,301,538 restitution order not cognizable under § 2255 because, as in Kaminski , payment obligation was limited to ten percent of defendant's monthly income, so as not to constitute custody); McEwan v. United States , 279 F.Supp.2d 462, 464 & n.1 (S.D.N.Y. 2003) (holding challenge to $6,984,000 restitution order not cognizable under § 2255 where defendant required to make payments in installments of fifteen percent of gross monthly income).
Here, the challenged judgments do not set terms for defendants' payment of restitution. Nevertheless, the record indicates that the district court plainly contemplated such terms, deferring its fixing of payment schedules until receipt of Probation Department recommendations, presumably upon defendants' release from prison when realistic assessments of their means can be made. This is evident from Lesniewski's and Baran's original judgments of conviction, which expressly state that the monetary parts of their sentences are to be paid "[a]s determined by probation." United States v. Ajemian , No. 11 Cr. 1091 (S.D.N.Y.), Dkt. Nos. 649, 704.3 While no *107similar statement is included in Rutigliano's or Ajemian's judgment of conviction, the omission appears to be scrivener's error because all four defendants' judgments of conviction state that the only monetary component of their sentences that is "due immediately" is each defendant's special assessment. Id. , Dkt. Nos. 401, 592, 649, 704. Moreover, all four judgments impose additional supervised release terms based on each defendant's "compliance with the installment payment schedule." Id. In sum, because defendants' original judgments of conviction plainly contemplate that restitution will be paid not immediately, but pursuant to terms yet to be set by the district court, and because there is no reason to think those terms will present the "rare situation" where restitution obligations so constrain liberty as to equate to custody, Gonzalez v. United States, 792 F.3d at 237, defendants' collateral challenges to the restitution parts of their sentences were not cognizable under § 2255.4
In seeking to avoid this conclusion, defendants argue that their restitution orders must be deemed custodial because they will never be able to make full restitution on any terms and, thus, necessarily face imprisonment. The argument fails because only a willful failure to pay restitution can result in incarceration, see 18 U.S.C. § 3614(b), and the law specifically prohibits incarceration solely on the basis of a defendant's financial inability to pay, see id. § 3614(c).
In sum, because defendants fail to show that the restitution ordered in their judgments of conviction equates to custodial punishment, the district court was without authority to review and reduce restitution pursuant to § 2255. See Kaminski v. United States , 339 F.3d at 87, 91.5
II. Baran's Alternative Pursuit of Coram Nobis Relief
Alone among the defendants, Baran pursued relief from restitution not only under § 2255, but also through a writ of error coram nobis pursuant to the All Writs Act. See 28 U.S.C. § 1651. The district court did not address this alternative ground for relief, and we do not direct it to do so on remand because we conclude that Baran cannot show fundamental error in her original restitution order, as required for coram nobis relief.
*108The writ of error coram nobis is an "extraordinary remedy" that "issues only in extreme cases," United States v. Denedo , 556 U.S. 904, 916, 129 S.Ct. 2213, 173 L.Ed.2d 1235 (2009), and "typically [is] available only when habeas relief is unwarranted because the petitioner is no longer in custody," Kovacs v. United States , 744 F.3d 44, 49 (2d Cir. 2014). The writ thus serves as "a remedy of last resort," Fleming v. United States , 146 F.3d 88, 89-90 (2d Cir. 1998), "strictly limited to those cases in which errors ... of the most fundamental character have rendered the proceeding itself irregular and invalid," Foont v. United States , 93 F.3d 76, 78 (2d Cir. 1996) (internal quotation marks omitted) (alteration in original).
To secure coram nobis relief, a petitioner must show that (1) "there are circumstances compelling such action to achieve justice," (2) "sound reasons exist for failure to seek appropriate earlier relief," and (3) "the petitioner continues to suffer legal consequences from his conviction that may be remedied by granting of the writ." Id. at 79 (internal quotation marks and alteration omitted); accord Kovacs v. United States , 744 F.3d at 49. The burden is a heavy one because a court reviewing a petition for coram nobis relief "must presume that the proceedings were correct, and the burden of showing otherwise rests on the petitioner." United States v. Mandanici , 205 F.3d 519, 524 (2d Cir. 2000).
The possibility that coram nobis could afford collateral relief from restitution was raised in our precedent by that part of the Kaminski opinion wherein Judge Calabresi wrote only for himself. See Kaminski v. United States , 339 F.3d at 89-91 ; see also Gonzalez v. United States , 792 F.3d at 238 n.31 (characterizing Kaminski 's discussion of coram nobis as dicta ); but see Barnickel v. United States , 113 F.3d 704, 706 (7th Cir. 1997) (recognizing availability of coram nobis for restitution challenges not cognizable under § 2255 ). In suggesting that restitution challenges not cognizable under § 2255 might be reviewed through coram nobis , Judge Calabresi was nevertheless careful to note that relief could be afforded "only if the challenged error is fundamental." Kaminski v. United States , 339 F.3d at 91. The challenged restitution orders here manifest no fundamental error. Thus, we need not decide if, or when, coram nobis might be invoked collaterally to challenge the restitution component of a criminal sentence because, even assuming Baran could do so here, her claim would necessarily fail on the merits.
Baran based her coram nobis claim-as she and her co-defendants based their § 2255 claims-on the fact that, after the RRB terminated disability pension benefits to annuitants whose claims had been supported by Ajemian or Lesniewski, the RRB approved benefits for a majority of those annuitants who reapplied. The district court was persuaded in part, concluding that the pre-termination benefits paid to reapproved applicants represented no loss to the RRB because it had made its own independent determination as to those applicants' disability statuses based on reliable information. See United States v. Ajemian , 2016 WL 6820730, at *3. Therefore, the district court reduced the challenged restitution orders to eliminate pre-termination benefits paid to reapproved applicants, explaining that "[t]he RRB may not seek reimbursement from the Defendants for disability payments it is required to make based on its own independent determinations." Id.
While the government challenges this conclusion, Baran (and her co-defendants) submit that it does not go far enough. They maintain that the district court should have vacated their restitution obligations *109entirely because the high rate of reapplication approvals leaves insufficient reliable evidence to conclude that any persons, including those who did not reapply for benefits, received disability pension benefits to which they were not entitled so as to cause a loss to the RRB.
We reject Baran's argument. Not only do the facts she relies on not show fundamental error in the original restitution order warranting wholesale vacatur; they fail to show fundamental error supporting any reduction of restitution.
In reaching that conclusion, we are guided by two general principles. First, the essence of fraud is misrepresentation, made with the intent to induce another person to take action "without the relevant facts necessary to make an informed ... decision." United States v. Binday , 804 F.3d 558, 579 (2d Cir. 2015). It is not necessary that the fraudster intend to inflict a financial loss. See id. Second, and by contrast, a court's power to order restitution is "limited to actual loss" and "may be awarded only in the amount of losses directly and proximately caused by [a] defendant's conduct." United States v. Gushlak, 728 F.3d 184, 194-95 (2d Cir. 2013) (emphasis omitted).
The record here, including the trial record supporting Baran's conviction, convincingly demonstrates that she and her co-defendants supplied the RRB with false medical documentation to induce its payment of many millions of dollars in disability pension benefits. There is no question that if the RRB had known the supporting medical documents were fraudulent, it would not have made these payments; thus, by making payments, the RRB sustained an actual loss. Cf. United States v. Marino , 654 F.3d 310, 322 (2d Cir. 2011) (observing, in rejecting causation challenge to restitution on direct appeal, that "but for" defendant's affirmative concealment of falsity of certain representations, victims would not have invested in venture "as no reasonable investor would invest in a known Ponzi scheme").
Baran and her co-defendants urge otherwise, arguing that the RRB sustained no loss in paying benefits to annuitants who could have demonstrated their entitlement to disability pensions without fraud. The argument fails. Nothing in the record shows that the RRB, once having been induced to pay disability benefits on the basis of fraudulent applications, remained obliged to pay those benefits to persons who purportedly could have demonstrated disability without fraud. Cf. United States v. Klein , 543 F.3d 206, 215 (5th Cir. 2008) (remanding for reduction of restitution where defendant showed insurance companies would have had to pay for medications actually provided to patients, "regardless" of defendant's fraudulent billing practices relating to how medications were administered). In any event, neither Baran nor her co-defendants offered evidence that any annuitants could have made a non-fraudulent showing of disability at the time of their original applications.
To be sure, at sentencing, the government bears the preponderance burden of proving actual loss supporting a restitution order. See United States v. Finazzo , 850 F.3d 94, 117 (2d Cir. 2017). But where, as here, the government carried that burden at the original sentencing by showing that the RRB's pre-termination payments were induced by fraud, a defendant seeking coram nobis relief cannot show fundamental error in the restitution order simply by speculating that some annuitants might have been able to demonstrate eligibility without fraud. See United States v. Mandanici , 205 F.3d at 524 (referencing coram nobis petitioner's burden to rebut presumption that underlying proceedings were correct);
*110Foont v. United States , 93 F.3d at 78 ("strictly limit[ing]" coram nobis relief to fundamental errors rendering original proceeding itself "irregular and invalid" (internal quotation marks omitted) ).
Defendants argue that more than speculation supports their challenges to the original restitution orders. Specifically, they assert that the RRB, in approving such a large number of disability pension reapplications based on reliable information, itself determined that it was required to pay pre-termination benefits to the approved re-applicants. The record does not support this conclusion.
After issuing Board Orders 13-33 and 13-55 terminating future disability pension payments to persons found disabled based on defendants' fraudulent submissions, the RRB allowed terminated annuitants to reapply for benefits. The review process for such reapplications, however, expressly stated that the RRB would not reopen or reevaluate an annuitant's original disability determination. Rather, reapplications would be assessed on the basis only of current medical evidence, specifically, evidence from within twelve months of the reapplication and not connected to Ajemian or Lesniewski. In short, reapplication approvals determined only an annuitant's present eligibility for disability pension benefits. In no way can these approvals be understood as "independent determinations" that reapplicants were disabled at the time of their initial-fraudulent-applications, much less that the RRB was "required" to pay these applicants the pre-termination benefits they received. United States v. Ajemian , 2016 WL 6820730, at *3.
Our conclusion is only reinforced by the temporal and factual gulf between the fraudulent applications that induced the RRB's payment of pre-termination benefits and the approved reapplications. An average 8.3-year interval separated the two. Moreover, some 80% of reapplications relied on at least one medical condition not included in the initial application to demonstrate present disability, while some 70% of reapplications omitted a medical condition asserted in the initial fraudulent applications. In these circumstances, an RRB finding on reapplication that a person was then eligible for a disability pension cannot be said to have found that he was also eligible years earlier or that the RRB considered itself "required to make" pre-termination benefits payments despite defendants' fraud.
In sum, the record convincingly supports the district court's initial determination of the actual loss caused to the RRB through its termination of benefits by defendants' fraud and, therefore, the restitution amounts ordered in defendants' original judgments. The RRB's subsequent approval of most reapplications for benefits shows no fundamental error in restitution because the RRB determined only a reapplicant's present eligibility for benefits; it made no determination as to eligibility in the pre-termination period.6 Thus, because Baran cannot show fundamental error in the $31,398,907 restitution ordered in her original judgment of conviction, she was not entitled to have that order vacated or even reduced through a writ of error coram nobis.
*111CONCLUSION
For the reasons stated, we conclude,
1. Collateral review jurisdiction under 28 U.S.C. § 2255 does not reach the restitution parts of criminal sentences where, as here, the restitution cannot be deemed custodial punishment. It is not the amount of restitution alone but, rather, the terms of payment that identify those rare cases in which a restitution order might so severely deprive a defendant of liberty as to equate to custodial punishment. Defendants' judgments of conviction reference payment schedules, but no such schedules have yet been set by the district court. Thus, there is no basis in the present record for equating defendants' restitution obligations to custodial punishments, without which the district court could not review collateral challenges to restitution pursuant to § 2255.
2. To the extent Baran also collaterally challenges restitution through a writ of error coram nobis , we need not decide whether coram nobis is available for that purpose because, even if it is, Baran cannot show fundamental error in the $31,398,907 restitution ordered in her original judgment of conviction. As a result, she is entitled neither to the vacatur of all restitution that she seeks nor to the reduction in restitution that the district court granted pursuant to § 2255.
Accordingly, we VACATE the district court's November 1, 2016 order reducing the restitution amounts ordered in defendants' original judgments of conviction, and we REMAND with directions to reinstate the original judgments and to dismiss all defendants' § 2255 petitions as well as Baran's coram nobis petition.