Opponents of Harvard College’s affirmative action policies asked the Supreme Court to bar colleges from using race as an admissions factor, setting up what could be a defining showdown for higher education and the court’s conservative majority.

The appeal by Students for Fair Admissions asks the court to overturn the 2003 ruling that let universities keep using race-conscious admissions to diversify their student bodies. A federal appeals court rejected the group’s lawsuit, which contends Harvard is violating federal civil rights laws by intentionally discriminating against Asian-Americans in undergraduate admissions.

The case arrives at a high court likely to be far more skeptical of race-based admissions than it was in 2003. Chief Justice John Roberts has repeatedly criticized what he sees as special protections for racial minorities, writing in a 2006 redistricting case, “It is a sordid business, this divvying up by race.”

The 2003 decision, Grutter v. Bollinger, upheld a University of Michigan Law School policy that considered race as part of a holistic look at a student’s application. The Grutter ruling came a quarter century after the court backed university affirmative action for the first time in the Bakke ruling, which upheld an earlier Harvard admissions policy.

The new appeal called the Grutter decision “grievously wrong” and said the decision had reinforced policies that discriminate against other historically oppressed minorities.

“Harvard is where this court’s approval of race-based admissions began, and it is a fitting place for that approval to end,” argued Students for Fair Admissions, a group run by longtime racial-preference foe Edward Blum.

In a statement, Harvard said it “remains deeply committed” to creating a diverse campus community.