RICHARD S. ARNOLD, Chief Judge.
Two appeals1 involving the Little Rock School District (LRSD) are consolidated for our review. In the first case, African-American voters2 appeal the District Court’s3 order dismissing their claim under the Voting Rights Act of 1965, § 2, as amended, 42 U.S.C. § 1973, against LRSD. In the second case, the Joshua Intervenors appeal the District Court’s August 10, 1994, order granting LRSD’s motion to close Ish Incentive School. We affirm in both cases.
I.
We review first Charles’s claim that the LRSD’s election-zone plan adopted by the Pulaski County Board of Education (PCBE) and approved by the District Court violates the Voting Rights Act. He alleges that the approved plan denies African-American voters within the LRSD equal opportunity to participate in the electoral process and to elect representatives of their choice. More specifically, he insists that the plan dilutes the vote of African-American residents of the LRSD because those residents are numerous and compact enough to justify the creation of three single-member zones in which they would be in the majority,4 instead of the two zones provided for in the adopted plan.5
In 1986, a plan was adopted containing seven single-member zones for election of members of the LRSD school board. Two of the zones were majority-minority zones.6 In July of 1992, Charles filed this complaint alleging that the 1986 plan should be revised on the basis of the 1990 census. Four rezoning plans were prepared and presented at a public meeting. The Pulaski County Board of Education (PCBE), the responsible body under Arkansas law, selected alternative four.7 Under alternative four, Zone Two is a majority-minority zone, and Zone One is a supermajority zone (65% or more minority population).
[909]*909On February 16, 1993, PCBE submitted the plan to the District Court for approval. Charles filed objections alleging that the plan violated § 2 of the Voting Rights Act. Following a trial, the District Court approved the plan adopted by the PCBE and dismissed Charles’s complaint. The Charles plaintiffs bring this appeal.
On several occasions we have reviewed challenges brought under the Voting Rights Act to Arkansas’s electoral practices. On each of these occasions we have set forth the requirements for establishing a violation of § 2 of the Voting Rights Act, and proceeded mindful of the fact that “equal political opportunity [is] the focus of the enquiry.” Johnson v. De Grandy, — U.S. -, -, 114 S.Ct. 2647, 2658, 129 L.Ed.2d 775 (1994). We do so once again in this case.
Section 2 of the Voting Rights Act of 1965, as amended in 1982, provides:
(a) No voting qualification or prerequisite to voting or standard, practice, or procedure shall be imposed or applied by any State or political subdivision in a manner which results in a denial or abridgement of the right of any citizen of the United States to vote on account of race or color, or in contravention of the guarantees set forth in section 1973b(f)(2) of this title, as provided in subsection (b) of this section.
(b) A violation of subsection (a) of this section is established if, based on the totality of the circumstances, it is shown that the political processes leading to nomination or election in the State or political subdivision are not equally open to participation by members of a class of citizens protected by subsection (a) of this section in that its members have less opportunity than other members of the electorate to participate in the political process and to elect representatives of their choice. The extent to which members of a protected class have been elected to office in the State or political subdivision is one circumstance which may be considered: Provided, That nothing in this section establishes a right to have members of a protected class elected in numbers equal to their proportion in the population.
42 U.S.C. § 1973. The leading case interpreting the statute is Thornburg v. Gingles, 478 U.S. 30, 106 S.Ct. 2752, 92 L.Ed.2d 25 (1986). In Gingles, the Supreme Court identified three necessary preconditions which must be established to maintain an action under § 2: (1) the minority group is sufficiently large and geographically compact to constitute a majority in one or more single-member districts; (2) the minority group is politically cohesive; and (3) the majority votes sufficiently as a block to enable it usually to defeat the minority’s preferred candidate. Id. at 50-51, 106 S.Ct. at 2766.
Once the plaintiff has established the three preconditions, a court must look to “the totality of the circumstances” to determine whether minority voters have less opportunity than other members of the electorate to participate in the political process and to elect representatives of their choice. Seven additional factors should be considered at this stage of the case:
1. the extent of any history of official discrimination in the state or political subdivision that touched the right of the members of the minority group to register, to vote, or otherwise to participate in the democratic process;
2. the extent to which voting in the elections of the state or political subdivision is racially polarized;
3. the extent to which the state or political subdivision has used unusually large election districts, majority vote requirements, anti-single shot provisions, or other voting practices or procedures that may enhance the opportunity for discrimination against the minority group;
4. if there is a candidate slating process, whether the members of the minority group have been denied access to that process;
5. the extent to which members of the minority group in the state or political [910]*910subdivision bear the effects of discrimination in such areas as education, employment and health, which hinder their ability to participate effectively in the political process;
6. whether political campaigns have been characterized by overt or subtle racial appeals;
7. the extent to which members of the minority group have been elected to public office in the jurisdiction.
S.Rep. No. 417, 97th Cong., 2d Sess. 28-30 (1982), reprinted in 1982 U.S.Code Cong, and Admin.News 177, 206-07.
Additional factors that in some cases have had probative value as a part of the plaintiffs’ evidence are:
1. whether there is a significant lack of responsiveness on the part of elected officials to the particularized needs of the members of the minority group;
2. whether the policy underlying the state or political subdivision’s use of such voting qualification, prerequisite to voting, or standard, practice or procedure is tenuous.
Id. at 30, 1982 U.S. Code Cong, and Admin. News at 207. This list is not comprehensive or exclusive. Gingles, 478 U.S. at 45, 106 S.Ct. at 2763.
The District Court began by holding that the relevant comparison is between the plan adopted by the PCBE and the 1986 plan, the one in effect right before the adoption of the challenged plan. With respect, we think this was not the correct standard. The statute asks whether minority voters have less political opportunity than other voters. The inquiry is whether minority opportunity is less under the challenged plan than what it would be under some other arrangement, one that would comply with § 2. Under the District Court’s approach, plaintiffs could never prevail unless the plan in suit was less favorable to them than whatever plan preceded it. They would have to prove “retrogression.” A majority of the Supreme Court has rejected that theory. “Retrogression is not the inquiry in § 2 dilution eases.” Holder v. Hall, — U.S. -, -, 114 S.Ct. 2581, 2587, 129 L.Ed.2d 687 (1994) (opinion of Kennedy, J., joined by Rehnquist, C.J.); id. at -, 114 S.Ct. at 2589 (O’Connor, J., concurring in part and concurring in the judgment) (the comparison must be with “an objectively reasonable alternative practice as a benchmark_”); id. at -, 114 S.Ct. at 2622 (Blackmun, J., dissenting, joined by Stevens, Souter, and Ginsburg, JJ.) (“minority voters’ potential ‘in the absence of the allegedly dilutive mechanism must be measured against the benchmark of an alternative structure or practice that is reasonable and workable under the facts of the specific case.”) (footnote omitted); accord, Jeffers v. Tucker, 847 F.Supp. 655, 658 (E.D.Ark.1994) (three-judge court).
In the present case, the best approach is to treat the plan proposed by the plaintiffs as a prima facie reasonable alternative for purposes of this case. We shall use this approach as a sort of working hypothesis. This is the framework used in Jeffers v. Tucker, supra. It makes sense because the plaintiffs’ plan is a way of making concrete their contention that the Gingles preconditions are met. Their plan shows, for example, in their view at least, that it is possible to create one more reasonably compact and contiguous district in which African-Americans will have a majority.
Two more preliminary points are in order.
First, Charles claims that § 2 requires the LRSD to maximize African-American representation. A similar claim was considered and rejected in Jeffers v. Tucker, supra. In the context of a challenge to state legislative districts in the same county where LRSD is located, the Jeffers v. Tucker Court concluded that § 2 did not require maximization of minority-group representation. See Jeffers, 847 F.Supp. at 657.8 After Jeffers v. [911]*911Tucker was decided, the Supreme Court spoke dispositively to this issue: “Failure to maximize cannot be the measure of § 2.” De Grandy, — U.S. at -, 114 S.Ct. at 2660.
Just as we reject Charles’s maximization claim, we must also reject LRSD’s claim that proportional representation precludes a finding of vote dilution under § 2. In De Grandy, supra, the Supreme Court considered the state of Florida’s argument that “no dilution occurs whenever the percentage of single-member districts in which minority voters form an effective majority mirrors the minority voters’ percentage of the relevant population.” Ibid. In rejecting this argument, the Court stated: “Proportionality ... would thus be a safe harbor for any districting scheme. The safety would be in derogation of the statutory text and its considered purpose, however, and of the ideal that the Voting Rights Act attempts to foster.” Ibid. Consequently, even if the LRSD can show proportional representation, the inquiry must not end there, but must encompass all factors relevant to the plaintiffs’ claim.
We assume for purposes of this appeal that plaintiffs have met the three Gin-gles preconditions: that they are sufficiently numerous and geographically compact to form a third majority-African-American district; 9 that voting in LRSD is racially polarized; and that white voters usually vote in sufficient numbers to defeat candidates preferred by African-Americans. See De Grandy, supra, — U.S. at -, 114 S.Ct. at 2656 (assuming without deciding that first condition has been satisfied). We proceed directly to analyze “the totality of the circumstances,” including the three Gingles preconditions, the “Senate factors,” and other relevant facts.
We turn first to the first two Senate factors, the ones said to be most important in Gingles, supra, 478 U.S. at 48-49 n. 15, 106 S.Ct. at 2765 n. 15. First, African-American candidates have enjoyed a reasonably good measure of success in Little Rock elections in general. The record contains evidence of two LRSD elections, one at large (conducted before state law was changed to require zone elections in districts of a certain size) and one in a zone. In both elections an African-American candidate defeated a white candidate.10 The District Court also considered 33 “exogenous” elections — races in Little Rock but not for the school board. African-American candidates won most of these races. Many of these victories can be explained by special circumstances, however (for example, an African-American running as an incumbent, or running in a plurality race against several white candidates). The second Senate factor, racially polarized voting, is without doubt present to a degree, as plaintiffs’ statistical proof has shown. There is a high correlation between the number of voters in a precinct and the number of votes cast for African-American candidates. On the other hand, there has been some decisive cross-over voting of whites for African-American candidates. See, e.g., Jeffers v. Clinton, 730 F.Supp. 196, 216 (E.D.Ark.1989) (three-judge court), aff'd mem., 498 U.S. 1019, 111 S.Ct. 662, 112 L.Ed.2d 656 (1991) (subsequent history omitted).
We discuss briefly the other factors. Certainly there is a history of official discrimination in Arkansas, though its present effects are less pronounced in Pulaski County than in some other parts of the State. There are no discriminatory voting practices now in use in the LRSD. There is a majority-vote requirement, but in elections by zone this prac-[912]*912tiee actually works to the advantage of African-American voters in zones where they have a majority. We do not think that the majority-vote requirement cuts strongly either way in this case. The socioeconomic status of African-Americans is lower than that of whites in the LRSD, and this condition is partly due to the effects of past racial discrimination. This factor cuts in favor of Charles, though, again, not so strongly as it would in the Delta region. The remaining factor — unresponsiveness to the concerns of minority citizens — has not been proved. The evidence on this subject is equivocal at best.
On the whole, we think these factors, when considered on the assumption that the three preconditions have been met, do not make a strong enough case in Charles’s favor. The judgment we have to make is necessarily imprecise. It is even, in some sense and to some degree, subjective. But our experience with these cases, especially Jeffers v. Tucker, supra, holding that § 2 does not require the creation of an additional majority-African-American State Senate district in Pulaski County, leads us to conclude that a violation of law has not been proved here. We find two aspects of the case especially persuasive. First, although the plaintiffs’ proposed zone boundaries are nowhere nearly so bizarre as the ones held presumptively unconstitutional by the Supreme Court in Shaw, they are markedly less regular and compact than those in LRSD’s adopted plan. And second, African-Americans have consistently achieved proportional representation on the LRSD board. Since 1983, they have continuously held two out of seven seats (28.6%). They make up 29% of LRSD’s voting-age population. This fact, while by no means an absolute defense, has recently been given special importance by the Supreme Court. See De Grandy, supra. Nor are we persuaded that this state of affairs is likely to change in the foreseeable future. On balance, we agree with the District Court that plaintiffs have not proved a violation of § 2 of the Voting Rights Act.
II.
Our second case involves the desegregation of Pulaski County schools. The Joshua Intervenors, representatives of the African-American students in Pulaski County, challenge the closing of Ish Incentive School. They maintain that it was an abuse of discretion for the District Court to close Ish without a hearing; that the closing of Ish is precluded by the settlement agreement; and that the Little Rock School District (LRSD) should be estopped to close Ish because it has failed to satisfy the settlement agreement’s recruitment requirements. Each of these arguments is considered below.
A.
In 1989, the parties to the desegregation ease — the LRSD, the Pulaski County Special School District (PCSSD), the North Little Rock School District (NLRSD), and the Joshua Intervenors, along with the McRnight Intervenors — entered into a settlement agreement. The agreement was approved by this Court in 1990. Little Rock School District v. Pulaski County Special School District, 921 F.2d 1371 (8th Cir.1990) (subsequent history omitted). This agreement put in place the framework for a concerted effort to end racial segregation in Pulaski County schools. The District Court has the onerous task of exercising jurisdiction over the implementation of the settlement agreement.
Under the agreement, four desegregation plans, one for each school district and an Interdistrict Plan, were formed. The LRSD and Interdistrict plans are the focus of this case. Under the LRSD Plan, the district was to establish eight incentive schools. These incentive schools would initially be all or virtually all African-American, with future projections of 50 to 62 per cent. African-American. The incentive schools would sponsor special compensatory-education programs and receive twice as much money per pupil as other elementary schools. Attendance at an incentive school would be open to African-American students living within the particular school’s attendance zone.
In the Interdistrict Plan, provisions were made for the establishment of six interdis-trict magnet schools in addition to the six interdistrict schools in existence when the plan was approved. These interdistrict schools were to emphasize distinct education[913]*913al themes. African-American students from the LRSD and white students from the PCSSD were expected to attend the interdis-trict schools.
Preparations are now under way for the building of King Interdistrict School. King’s future location was designated in the Inter-district Plan to be “in downtown Little Rock ... in the general area along 1-630 between 1-30 and University Avenue.” LRSD Plan, at 139. The District Court approved the specific site for King on March 17, 1992. Leonard Thalmueller was hired to design King’s attendance zone. Dr. Thalmueller prepared three plans for review by the Office of Desegregation, which presented the plans to the LRSD. In the plan adopted by LRSD, the King attendance zones included Ish Incentive School’s attendance zones.
On May 5,1993, LRSD moved for approval of King’s attendance zones. Over the Joshua Intervenors’ objections, the District Court gave conditional approval to the King attendance zones on June 11, 1993. The District Court’s approval was conditioned on Ish’s remaining open unless the LRSD established that fewer than 100 students wanted to attend Ish during the 1993-94 academic year. During the 1992-93 academic year, 229 students lived in Ish’s attendance zones. Ish had an enrollment of 183 students. Ninety-three were from within the Ish attendance zone, while the other 90 resided outside its attendance zone. The conditional approval required that LRSD survey students within the Ish attendance zone, including those not attending Ish currently, and all students attending Ish, including students from outside the Ish attendance zone.
On June 11, 1993, the District Court entered an order requiring LRSD to submit the proposed survey, along with a plan for executing the survey and implementing the survey results, within ten days. The court gave Joshua five days to file responses and objections to the proposed survey and plan. No objections were filed. In compliance with the District Court’s order, LRSD filed its proposed survey and its plan for implementing the survey on June 21,1993. The district proposed sending form letters, King and Ish fact sheets, and a School Selection Form to parents of potential students. Accordingly, the District Court approved the survey and implementation plan on June 30, 1993.
' The survey was then conducted. LRSD mailed 260 survey forms, and 173 of the forms were returned. Of the 173 forms returned, 82 students requested Ish. Based on the survey results, LRSD moved the District Court to close Ish. Joshua filed a response to the motion to close Ish expressing opposition, challenging the survey process, and requesting a hearing. On August 2, 1993, the Court denied Joshua’s objections as untimely, and granted LRSD’s motion to close Ish.
B.
First, we address Joshua’s claim that remand for a hearing is required. While we recognize the possible validity of some of the objections raised by Joshua, we cannot say that the District Court acted arbitrarily by refusing to hold a hearing. In its order dated June 11, 1993, the Court specifically requested that objections to the survey be submitted within five days after LRSD filed the proposed survey and implementation plan. LRSD’s proposed survey and plan were filed on June 21,1993. Joshua made no objections to the survey format or the implementation plan until the survey was completed, more than a month later. Most of the objections Joshua makes now could have been made prior to the survey. It was important for the matter to be decided quickly. The District Court’s decision to expedite its ruling concerning Ish’s future by providing a timeline for pleadings was not arbitrary. The Court did not abuse its discretion.
C.
Turning to the heart of this appeal, we now consider the substance of the decision to modify the settlement agreement and approve the closing of Ish Incentive School. The District Court gave two reasons for its decision. First, the survey results indicated that fewer than 100 students wanted to attend Ish during the 1993-94 academic year. Second, the District Court cited the opening of King Interdistrict School as a desegregated alternative to Ish.
[914]*914A party seeking modification of a consent decree “must establish that a significant change in facts or law warrants revision of the decree.” Rufo v. Inmates of Suffolk County Jail, 502 U.S. 367, 393, 112 S.Ct. 748, 764, 116 L.Ed.2d 867 (1992). If the moving party meets this burden, the District Court must then determine “whether the proposed modification is suitably tailored to the changed circumstance.” Id. at 391, 112 S.Ct. at 763. The modification “must not create or perpetuate a constitutional violation,” nor “strive to rewrite a consent decree so that it conforms to the constitutional floor.” Ibid.
In this case, Joshua insists that the settlement agreement requires that the district maintain incentive schools and provide double funding for them. Closing Ish, she maintains, is in direct conflict with the settlement agreement. In addition, she asserts that the reasons espoused by the Court for closing Ish do not provide a proper basis for a major modification of .the settlement agreement. But nothing in the settlement agreement requires the district to maintain a particular incentive school for an indefinite period of time. What the agreement requires is that the district maintain a number of incentive schools “sufficient to accommodate that number of black students who, by attending those schools, make it possible to achieve a student population in the remaining Little Rock Schools ... of 55 percent black and 45 percent white with a variance of 5 percent.” Interdistrict Plan, April 29, 1992, at 4. At the outset the parties contemplated that “[a]s new Interdistrict Schools are established those seats attributable to LRSD will be available for those students who otherwise would or could have been assigned to an incentive school.” Ibid.
Several factors persuade us that the District Court’s action was permissible. First, the parents of students eligible to attend seem to prefer to send their children elsewhere when given a choice. The survey results evidence low community interest in maintaining Ish as an incentive school. Second, Ish students fall within the attendance zone of the new King Interdistrict School and may choose to attend King. King, like Ish, will offer program enhancements. King’s program enhancements include four curriculum specialists, a 56-station computer lab, electronie-assisted instruction, and an automated media center. Moreover, King is in close proximity to Ish, it is located in an African-American neighborhood, and perhaps most importantly, King will be integrated.
We stated early on that the passage of time would necessitate modifications in the desegregation plans. Appeal of Little Rock School Dist., 949 F.2d 253, 254 (8th Cir.1991). A designated goal of the interdistrict schools was to achieve desegregation by reducing the number of African-American students in the LRSD while reducing the number of white students in the PCSSD. See Little Rock School District, 921 F.2d at 1379. We are convinced that the closing of Ish advances this goal by offering a desegregated setting for students within the Ish attendance zone alongside children from the PCSSD.
What is important here is that students be provided the opportunity to gain an education in a desegregated environment. Indeed, the purpose underlying double funding is to compensate African-American students attending incentive schools for the repercussions of racial isolation due to past and present discrimination. The students eligible to attend Ish may now avail themselves of a desegregated school if they choose. In their case, double funding will not be required because they no longer face racial segregation within their school.
D.
Joshua also advances an estoppel argument. In essence, Joshua contends that the LRSD should be barred from closing Ish because Ish’s low attendance is due to LRSD’s failure to implement its recruitment duties required by the settlement. This argument has rudimentary appeal, particularly in light of the District Court’s observations regarding LRSD’s lackluster efforts, or lack thereof, at recruiting students to attend Ish. We think this argument, however, clouds the real issue — whether closing Ish advances the goals of the desegregation plan. Our concern here is with achieving this goal, not with [915]*915punishing the LRSD for recruitment improprieties.
We have recognized that “estoppel is an equitable doctrine, and it should not be given effect beyond what is necessary to accomplish justice between the parties.” Maitland v. University of Minnesota, 43 F.3d 357, 364 (8th Cir.1994). Justice would not be served by requiring Ish to remain open when the evidence indicates that the goal of desegregation will be served by closing it. Thus, Joshua’s estoppel argument must fail.
III.
We affirm the order dismissing Charles’s voting-rights claim. We also affirm the order approving the closing of Ish Incentive School.
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