JERRY E. SMITH, Circuit Judge:
Elzie Ball, Nathaniel Code, and James Magee are death row inmates in the Louisiana State Penitentiary (“LSP”) and are housed in cells without air conditioning. The three sued in 2013, claiming a violation of the Eighth Amendment. Their case comes to us for the second time, after a different panel found that an Eighth Amendment violation had occurred and that injunctive relief was appropriate but that the district court had exceeded the bounds of the Prison Litigation Reform Act (“PLRA”) and Gates v. Cook, 376 F.3d 323, 339-40 (5th Cir. 2004), by mandating facility-wide air conditioning and setting a maximum heat index. See Ball v. LeBlanc, 792 F.3d 584, 596, 598-600 (5th Cir. 2015) (“Ball I”). Because the district court did not adhere to the mandate, we reverse and remand.
I.
A.
The basis of the complaint is that plaintiffs have pre-existing medical conditions that render them vulnerable to heat-related injury. A detailed description of the death-row facility, located in Angola, Louisiana, can be found in Ball I, id. at 589-91. Most relevant here, the cells are without air conditioning, which has resulted in heat indices of over 100 degrees. Moreover, before suing, plaintiffs had only limited access to ice and could take only hot showers. The Ball I panel agreed with the finding of a constitutional violation: “[W]e affirm the district court’s conclusion that housing these prisoners in' very hot cells without sufficient access to heat-relief measures, while knowing that each suffers from conditions that render him extremely vulnerable to serious heat-related injury, violates the Eighth Amendment.” Id. at 596.
The Ball I panel also concluded, however, that the initial injunction (the “First Plan”) violated the PLRA. Id. at 598-600. Under the First Plan, the court effectively required the state “to install air conditioning throughout death row housing” by developing “a plan to reduce and maintain the heat index in the Angola death row tiers at or below 88 degrees Fahrenheit.” Id. at 598 (quoting Ball v. LeBlanc, 988 F.Supp.2d 639, 698 (M.D. La. 2013)). “The PLRA greatly limits a court’s ability to fashion injunctive relief.” Id. Courts may order only relief that “extends no further than necessary to correct the violation of the Federal right, and is the least intrusive means necessary to correct the violation.” Id. (quoting 18 U.S.C. § 3626(a)(1)(A)).
Accordingly, the First Plan violated the PLRA, in part1 because air conditioning was “unnecessary to correct the Eighth Amendment violation.” Id. at 599.2 The panel suggested “acceptable remedies short of facility-wide air conditioning,” such as (1) diverting “cool air from the guards’ pod into the tiers,” (2) allowing access to air conditioned areas during tier time, (3) allowing “access to cool showers at least once a day,” (4) giving “ample” cold drinking water and ice “at all times,” (5) providing “personal ice containers and individual fans,” and (6) installing “additional ice machines.” Id. The panel told the district court to “limit its relief to these types of remedies.” Id.
Additionally, the relief required under the First Plan was far broader than that approved of in Gates. Id. at 600. “The Gates court did not mandate a maximum heat index.... It required particular heat measures, including fans, ice water, and showers, ‘if the heat index reaches 90 degrees or above.’ ” Id. (quoting Gates, 376 F.3d at 336). The panel noted that the First Plan required relief that was far more extensive and expensive than what Gates allowed and that because “Gates upheld an injunction providing narrower relief, and there is no showing that the Constitution mandated more relief for these prisoners for the same prison condition in this case, on remand the court must craft relief more closely aligned with Gates as well as consistent with the PLRA.” Id.
B.
On remand, the district court ordered the state to submit a new plan in light of this court’s mandate, whereupon the state submitted its “ ‘Second Heat Remediation Plan’ or ‘Second Plan.’ ” That plan provided that plaintiffs would have cold water for their daily, fifteen-minute showers; it gave each plaintiff ice containers that would be regularly replenished fi’om newly purchased ice machines; and it provided each plaintiff with a personal fan. Unsatisfied, plaintiffs moved to modify, urging the court to reinstate its initial plan—ie., the very plan that Ball I had explicitly rejected.
In connection with simultaneous settlement discussions, the state implemented additional, experimental relief measures, consistent with the stipulation that “any discussions or actions taken would not be admissible as evidence in this case pursuant to ... Federal Rule of Evidence 408(a)(2).” These exploratory remedies, which the court termed the “Third Plan,” are the basis for the later additional relief mandated by the modified second injunction at issue on this appeal. Moreover, the Special Master informed the parties that the district court had “advised that the implementation of any efforts or measures, on a trial basis, in this case will not be viewed as spoliation or destruction of evidence.... [Tjhese discussions are confidential and will remain so as long as the parties so request.”
The court then held two hearings. At the first, it heard evidence from Dr. Vassallo, who had testified in the initial trial and substantially reiterated her testimony. Additionally, each of the plaintiffs testified that, even after the implementation of the Second Plan, they experienced the same heat-related symptoms as before; At the second hearing, the Special Master testified about the Third Plan. Although the state objected that such evidence was inadmissible-under. Federal Rule of Evidence 408, the district court reasoned that it would not require disclosure of “any communications among the parties” but that it had to learn about the changes in plaintiffs’ conditions of confinement, which relate to a constitutional violation. Accordingly, the court overruled the objection and admitted evidence of the Third Plan.
The court issued an injunction in accordance with the Third Plan, reasoning that the Second Plan did not reduce the substantial risk of serious harm because the plaintiffs .continued to experience heat-related symptoms even during its implementation. Ball, 223 F.Supp.3d at 529, 545, 554-57. The court believed that “the only means to reduce the substantial risk of serious harm to Plaintiffs, and thereby remedy the Eighth Amendment violation in this case, is to lower the' temperatures and heat indices to which Plaintiffs are. exposed.” Id. at 545.
Accordingly, the district court imposed the Third Plan, which contained the same ■ requirements as the Second Plan but also.
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JERRY E. SMITH, Circuit Judge:
Elzie Ball, Nathaniel Code, and James Magee are death row inmates in the Louisiana State Penitentiary (“LSP”) and are housed in cells without air conditioning. The three sued in 2013, claiming a violation of the Eighth Amendment. Their case comes to us for the second time, after a different panel found that an Eighth Amendment violation had occurred and that injunctive relief was appropriate but that the district court had exceeded the bounds of the Prison Litigation Reform Act (“PLRA”) and Gates v. Cook, 376 F.3d 323, 339-40 (5th Cir. 2004), by mandating facility-wide air conditioning and setting a maximum heat index. See Ball v. LeBlanc, 792 F.3d 584, 596, 598-600 (5th Cir. 2015) (“Ball I”). Because the district court did not adhere to the mandate, we reverse and remand.
I.
A.
The basis of the complaint is that plaintiffs have pre-existing medical conditions that render them vulnerable to heat-related injury. A detailed description of the death-row facility, located in Angola, Louisiana, can be found in Ball I, id. at 589-91. Most relevant here, the cells are without air conditioning, which has resulted in heat indices of over 100 degrees. Moreover, before suing, plaintiffs had only limited access to ice and could take only hot showers. The Ball I panel agreed with the finding of a constitutional violation: “[W]e affirm the district court’s conclusion that housing these prisoners in' very hot cells without sufficient access to heat-relief measures, while knowing that each suffers from conditions that render him extremely vulnerable to serious heat-related injury, violates the Eighth Amendment.” Id. at 596.
The Ball I panel also concluded, however, that the initial injunction (the “First Plan”) violated the PLRA. Id. at 598-600. Under the First Plan, the court effectively required the state “to install air conditioning throughout death row housing” by developing “a plan to reduce and maintain the heat index in the Angola death row tiers at or below 88 degrees Fahrenheit.” Id. at 598 (quoting Ball v. LeBlanc, 988 F.Supp.2d 639, 698 (M.D. La. 2013)). “The PLRA greatly limits a court’s ability to fashion injunctive relief.” Id. Courts may order only relief that “extends no further than necessary to correct the violation of the Federal right, and is the least intrusive means necessary to correct the violation.” Id. (quoting 18 U.S.C. § 3626(a)(1)(A)).
Accordingly, the First Plan violated the PLRA, in part1 because air conditioning was “unnecessary to correct the Eighth Amendment violation.” Id. at 599.2 The panel suggested “acceptable remedies short of facility-wide air conditioning,” such as (1) diverting “cool air from the guards’ pod into the tiers,” (2) allowing access to air conditioned areas during tier time, (3) allowing “access to cool showers at least once a day,” (4) giving “ample” cold drinking water and ice “at all times,” (5) providing “personal ice containers and individual fans,” and (6) installing “additional ice machines.” Id. The panel told the district court to “limit its relief to these types of remedies.” Id.
Additionally, the relief required under the First Plan was far broader than that approved of in Gates. Id. at 600. “The Gates court did not mandate a maximum heat index.... It required particular heat measures, including fans, ice water, and showers, ‘if the heat index reaches 90 degrees or above.’ ” Id. (quoting Gates, 376 F.3d at 336). The panel noted that the First Plan required relief that was far more extensive and expensive than what Gates allowed and that because “Gates upheld an injunction providing narrower relief, and there is no showing that the Constitution mandated more relief for these prisoners for the same prison condition in this case, on remand the court must craft relief more closely aligned with Gates as well as consistent with the PLRA.” Id.
B.
On remand, the district court ordered the state to submit a new plan in light of this court’s mandate, whereupon the state submitted its “ ‘Second Heat Remediation Plan’ or ‘Second Plan.’ ” That plan provided that plaintiffs would have cold water for their daily, fifteen-minute showers; it gave each plaintiff ice containers that would be regularly replenished fi’om newly purchased ice machines; and it provided each plaintiff with a personal fan. Unsatisfied, plaintiffs moved to modify, urging the court to reinstate its initial plan—ie., the very plan that Ball I had explicitly rejected.
In connection with simultaneous settlement discussions, the state implemented additional, experimental relief measures, consistent with the stipulation that “any discussions or actions taken would not be admissible as evidence in this case pursuant to ... Federal Rule of Evidence 408(a)(2).” These exploratory remedies, which the court termed the “Third Plan,” are the basis for the later additional relief mandated by the modified second injunction at issue on this appeal. Moreover, the Special Master informed the parties that the district court had “advised that the implementation of any efforts or measures, on a trial basis, in this case will not be viewed as spoliation or destruction of evidence.... [Tjhese discussions are confidential and will remain so as long as the parties so request.”
The court then held two hearings. At the first, it heard evidence from Dr. Vassallo, who had testified in the initial trial and substantially reiterated her testimony. Additionally, each of the plaintiffs testified that, even after the implementation of the Second Plan, they experienced the same heat-related symptoms as before; At the second hearing, the Special Master testified about the Third Plan. Although the state objected that such evidence was inadmissible-under. Federal Rule of Evidence 408, the district court reasoned that it would not require disclosure of “any communications among the parties” but that it had to learn about the changes in plaintiffs’ conditions of confinement, which relate to a constitutional violation. Accordingly, the court overruled the objection and admitted evidence of the Third Plan.
The court issued an injunction in accordance with the Third Plan, reasoning that the Second Plan did not reduce the substantial risk of serious harm because the plaintiffs .continued to experience heat-related symptoms even during its implementation. Ball, 223 F.Supp.3d at 529, 545, 554-57. The court believed that “the only means to reduce the substantial risk of serious harm to Plaintiffs, and thereby remedy the Eighth Amendment violation in this case, is to lower the' temperatures and heat indices to which Plaintiffs are. exposed.” Id. at 545.
Accordingly, the district court imposed the Third Plan, which contained the same ■ requirements as the Second Plan but also. required the state to (1) relocate plaintiffs to another tier, close to the guards’ pod, (2) install an air vent in the guards’ pod to divert cool air to plaintiffs’ cells, (3) set up a plastic curtain around plaintiffs’ cells to trap the cool air, (4) provide each plaintiff with an “IcyBreeze” unit, which is essentially an ice chest that blows cold air, and (5) regularly replenish the IcyBreeze units with ice. Id. The injunction would take effect only when the heat index exceeds 88 degrees. Moreover, the court provided that “[i]n the event that' mold growth proliferates in the guards’ pod” caused by the Third Plan, the state'is enjoined “to seal the air vent and provide a sufficient number of additional IcyBreeze units to each plaintiff in order to maintain the heat index” to “below 88 degrees Fahrenheit.” Id. at 548. The court concluded that those measures would sufficiently lower “the in-dices to which Plaintiffs are exposed” to “below the 88-degree benchmark.” Id.
The state appealed, contending that the district court had violated the Ball I mandate by (1) ordering a maximum heat index and (2) requiring air conditioning in the form of IcyBreeze machines. The state also maintains that the court violated Federal Rule of Evidence. 408 by introducing evidence of the Third Plan.
II.
“We review de novo a district court’s application of [a] remand order, including whether the law-of-the-case doctrine or mandate rule forecloses the district court’s actions -on remand.” United States v. Teel, 691 F.3d 578, 583 (5th Cir. 2012) (citation omitted). In their briefs and at oral argument, plaintiffs insist that we should review the modified injunction for abuse of discretion. Although modifications of injunctions are typically reviewed for abuse of discretion, see Baum v. Blue Moon Ventures, LLG, 513 F.3d 181, 190 (5th Cir. 2008), the .issue here is whether that modification was barred by Ball Ts mandate.3 Accordingly, we review de novo whether the modified injunction violates the mandate rule or the law-of-the-case doctrine.4
“Under the law-of-the-case doctrine, an issue of fact or law decided on appeal may not be reexamined either by the district court on remand or by the appellate- court on a subsequent appeal.” United States v. Carales-Villalta, 617 F.3d 342, 344 (5th Cir. 2010) (internal citation omitted). “The mandate rule is but a corollary to the law of the case doctrine.” United States v. McCrimmon, 443 F.3d 454, 460 (5th Cir. 2006)., Both give, way to three exceptions: “(1) [T]he evidence at a subsequent trial is substantially different; (2) there has been an intervening change of law by a controlling authority; (3) the earlier decision is clearly erroneous and would work a manifest injustice.” Id.; Gene & Gene, L.L.C. v. BioPay, L.L.C., 624 F.3d 698, 702 (5th Cir. 2010). On remand, a district court must implement “both the letter and the spirit” of the panel’s mandate. McCrimmon, 443 F.3d at 459.
Plaintiffs suggest that the new-evidence exception applies to override the mandate rule. We disagree. The only new evidence plaintiffs can point to is the Third Plan itself and allegedly new scientific,testimony. But' all of the relevant testimony— especially Vassallo’s critical testimony— was materially unchanged.5 And the evidence of the Third Plan itself could be relevant only in that it proved the feasibility of the Third Plan. Because the state’s claims center on. the propriety of a maximum heat index and the potential for air conditioning, such evidence would be irrelevant to our application of the mandate rule.6 Accordingly, no exception to that rule applies.
The state’s primary claim is that the district court violated the mandate rule by effectively requiring a maximum heat index. According to the state, Ball I foreclosed relitigating whether the Constitution required setting a maximum heat index. And, the state contends, the district court misapplied Ball I by finding that a maximum heat index.was necessary to remedy the constitutional violation. Cf. Nat’l Airlines, 430 F.2d at 960. Plaintiffs reply only that the specific measures required by the district court were blessed in Ball I.
We agree with the state. Ball I plainly foreclosed any consideration of a maximum heat index. As that panel explained, “The Gates court did not mandate a maximum heat index,” and the district court had to “limit its relief’ to the kinds of measures found in Gates. Ball I, 792 F.3d at 599-600.7 Although well-intentioned, the district court, to the contrary, both considered and accepted the need for a maximum heat index,8
Relying on a maximum heat index of 88 degrees, the court concluded that the Second Plan was inadequate because it exposed plaintiffs to heat indices above that. 'And based on that same maximum, the court adopted the Third Plan because it would lower the heat indices to below 88 degrees.9 Moreover, the court gave a provisional order regarding the possibility of mold growth—in that event, the state would have to provide enough IcyBreeze units to keep the heat index below 88 degrees. Therefore, the court violated the mandate by incorporating a maximum heat index into its order. Based on that violation, we reverse and remand the injunction.
On remand, the district court must reevaluate the necessity of the Third Plan even without a maximum heat index. It may well be that parts of the Third Plan are still necessary to redress the constitutional violation: ie., “housing these prisoners in very hot cells without sufficient access to heat-relief measures.” Id. at 596 (emphasis added).10 But the court cannot decree whether any given plan is necessary to lower the heat index to below a maximum, nor can it require the state to provide an undetermined number of Icy-Breeze units or other measures to keep the heat index below a certain point.
The district court must ensure that any relief “is narrowly drawn, extends no further than necessary to correct the violation of the Federal right, and is the least intrusive means necessary to correct the violation.” Id. at 598 (quoting 18 U.S.C. § 3626(a)(1)(A)). And although the court is not limited to the specific relief approved of by Gates, it may not order measures that are more extensive or intrusive than was the relief in Gates. See id.
C.
It will help the district court and the parties for us to examine whether some of the specific measures required by the Third Plan exceed the Ball I mandate. The state suggests that the IcyBreeze machines might be construed as air conditioning—which Ball I, id. at 599, expressly forbade. The state posits that IcyBreeze machines are functionally much like air conditioning in that they produce cold air.
We disagree. As the district court rightly explained, the IcyBreeze units are basically ice chests with fans attached. The chest blows out cool air but does not emit water vapor. In short, they are similar to evaporative coolers. And Ball I specifically approved of remedial measures such as ice chests and fans. Id. More importantly, Icy-Breeze machines are compact and inexpensive, each costing just over five hundred dollars.11 They therefore fit comfortably within Ball 7’s admonition that any relief must not be unduly intrusive and must take into account “any adverse impact on public safety or the operation of a criminal justice system.” Id. at 598-99 (quoting 18 U.S.C. § 3626(a)(1)(A)).
The rest of the injunction does not exceed the Ball I mandate. For example, the Ball I court specifically approved of requiring ice, cold showers, and fans. Id. at 599. And it allowed diverting cool air from the guards’ pod—provided, of course, that “such relief is narrowly drawn, extends no further than necessary to correct the violation of the Federal right, and is the least intrusive means necessary to correct the violation.” Id. at 598 (quoting 18 U.S.C. § 3626(a)(1)(A)). Moreover, Ball I approved the use of a temperature trigger. Id. at 600 (explaining that Gates “required particular heat measures ... if the heat index reaches 90 degrees or above”) (internal quotations omitted).
Indeed, a temperature trigger is necessary- to ensure that the injunction is inapplicable “during months when there is no heat risk to the Plaintiffs.” Id. It is by effectively requiring a temperature ceiling that the district court went astray. Accordingly, despite that we reverse based on the erroneous adoption of a maximum heat index, we leave open the possibility that, on remand, the court may require Icy-Breeze units or temperature triggers.
III.
The state posits that the district court admitted evidence of the Third Plan in violation of Federal Rule of Evidence 408.12 Evidentiary rulings are reviewed for abuse of discretion. Id. at 591 (citing Battle ex rel. Battle v. Mem’l Hosp. at Gulfport, 228 F.3d 544, 550 (5th Cir. 2000)). Moreover, “[e]ven if the court abused its discretion, this court will presume the error is harmless.” Id. (citation omitted). “The party asserting the error has the burden, of proving that the error was prejudicial.” Id. (citation omitted). The state asserts that the court violated Rule 408 because the Third Plan was regarding conduct during compromise negotiations and was a subsequent remedial measure.
Rule 408 precludes admitting any “conduct or ... statement made during compromise negotiations about the claim” “to prove or disprove the validity or amiount of a disputed claim or to impeach by a prior inconsistent statement or a contradiction.” FED. R. EVID. 408(a)(2). Moreover, the parties stipulated that “any discussions or actions taken [with regard to the Third Plan] would not be admissible as evidence in this case pursuant to ... Federal Rule of Evidence 408(a)(2).” The district judge even communicated, through the Special Master, that “the implementation of any efforts or measures, on a trial basis,” would hot be “viewed as spoliation or destruction of evidence” and that the discussions would remain confidential so long as the parties so requested. Accordingly, the Third Plan and any accompanying- discussions -were “conduct” and' “statement[s] made during compromise negotiations.” FED. R. EVID. 408(a)(2).
Yet Rule 408 contains a broad exception: “The court may admit this evidence for-another purpose, such as proving a witness’s bias, or prejudice, negating a contention of undue. delay, or proving an effort to obstruct a criminal investigation or prosecution.” FED. R. EVID. 408(b). Here, the other purpose relates to the court’s ongoing supervisory power over its injunction. See Plata, 563 U.S. at 542-45, 131 S.Ct. 1910. The district court may have been concerned about how the Third Plan would affect the prisoners and their constitutional rights; the court also could have wanted to know whether the Third Plan was more efficient than the Second Plan.13
Moreover, “Rule 408 should not exclude more than required to effectuate its goals, which, after all, run counter to the overarching policy favoring admission of all relevant evidence.” Lyondell Chem. Co. v. Occidental Chem. Corp., 608 F.3d 284, 299 (5th Cir. 2010). Courts must ensure that Rule 408 remains “tethered to the rationales underlying the rule.” Id. at 298. Those rationales—the irrelevancy of such evidence and the public policy in favor of compromise—are inapplicable here.14 The evidence was relevant and probative. And the public policy in favor of compromise could be satisfied' by excluding any negotiations 15—which would have formed the basis for the state to move to modify the injunction or enter a consent decree—but still admitting conduct affecting the prisoners’ constitutional rights (a manifest public policy concern, see Plata, 563 U.S. at 510-11, 542-43, 131 S.Ct. 1910). Accordingly, the district court did not abuse its discretion by admitting evidence of the Third Plan itself.16
Because the district court' erroneously ■addressed the propriety of a maximum heat index, found that it was: necessary, and issued a modified injunction that in certain instances incorporated it, the order imposing the modified injunction is REVERSED and REMANDED. We are confident that, on remand, the district court will conscientiously proceed in a manner that is consistent with this opinion and Ball 1.