The 20 New Haven firefighters who sued the city for discrimination in 2004 filed a draft order in federal court Friday asking for most of them to be promoted.

The firefighters argue that 15 of them should be promoted because they achieved the requisite scores in a disputed 2003 exam. Under that test, no minorities qualified for promotion, so the city threw out the results. Last summer, the Supreme Court ruled the city’s decision unconstitutional.

The city of New Haven filed its own order Friday, arguing that the city should promote certain firefighters who scored well in the 2003 exam, whether they were plaintiffs in the case or not.

The city’s draft order did not include names, but it said city officials would “prepare an eligible list” of firefighters to be promoted to lieutenant and captain.

The New Haven Fire Department has not promoted an officer in the six years, since the disputed test.

City Corporation Counsel Victor Bolden has said repeatedly that the city wants to move on from the lawsuit. In a statement released Friday, he said, “We look forward to proceeding with promotions and end this aspect of litigation.”

The legal dispute started in 2004, when the city threw out the results of the promotion test because most of the high scorers were white. The city argued it would be vulnerable to litigation from minority firefighters if it promoted officers who were almost all white. Now known as the “New Haven 20,” 19 white firefighters and one Hispanic sued the city for discrimination that year.

In June, the U.S. Supreme Court sided with the firefighters and ruled that the city had violated the 1964 Civil Rights Act. The Supreme Court then sent it back down to lower federal court to determine which firefighters should be promoted and what compensation they should receive.

The city is also trying to delay two lawsuits filed by black firefighters, including Michael Briscoe, who filed a federal lawsuit against city officials last month, claiming that the same 2003 exam discriminated against blacks and unjustly denied him promotion to lieutenant. On Nov. 2, the city motioned to prevent black firefighter Michael Briscoe’s lawsuit from proceeding.

If Briscoe’s lawsuit proceeds, the city may not be able to certify the results of the 2003 test and promote its officers.