(CNN) A lawsuit that claims Harvard caps the number of high achieving Asian-Americans it admits could go to trial in Boston as early as this summer, according to a new filing in the case.

The lawsuit, begun in 2014 by a conservative advocate who has long challenged affirmative action that benefits blacks and other historically disadvantaged racial minorities, could affect who gets a place on the selective Ivy League campus.

It could have even greater consequences nationwide. The case was devised ultimately to topple a 1978 US Supreme Court decision that first endorsed college affirmative action, Regents of the University of California v. Bakke.

Lawyers for both sides submitted a status report and proposed schedule Friday in US District Court in Boston. Harvard has asked that a trial begin in July or August. Students for Fair Admissions, the group that filed the complaint, wants a trial to begin no earlier than October 1. It says it would need the extra time to prepare for a trial of the size and scope anticipated.

The Students for Fair Admissions challengers, however, also say they want to file a summary-judgment motion by June 15, which would test whether they have already made the case that Harvard intentionally discriminates against Asian-Americans. Harvard contends that the group lacks the grounds even to make the motion and urged US District Judge Allison Burroughs to move directly to a trial.

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