CNN  — 

Lawyers for President Donald Trump said Wednesday they will raise additional challenges to the Manhattan district attorney’s subpoena for his financial records after the Supreme Court ruled a president does not have broad immunity against a state grand jury investigation.

In a joint filing made with the lower court judge, lawyers for Trump cited a concurring opinion written by Supreme Court Justice Brett Kavanaugh, and joined by Neil Gorsuch, that said the President “may raise further arguments as appropriate,” including whether the subpoena is too broad and if it would impede the President’s ability to carry out his job.

“The President intends to raise some or all of these arguments in his forthcoming Second Amended Complaint,” Trump’s lawyers wrote.

Last week, the Supreme Court ruled that the President was not immune from a state grand jury investigation, dealing a victory to the Manhattan District Attorney Cyrus Vance. His office is investigating whether the President and his company violated state law relating to the hush-money payments made to women who alleged affairs with the President and their reimbursement to former fixer Michael Cohen.

Trump argued in the brief that the district court must wait to act until the Supreme Court has entered a judgment in the case, but Vance’s office disputed that notion. “[T]he Second Circuit never lost its power to act in this matter,” Vance’s office wrote. “And the Second Circuit’s mandate remanding the matter to this Court in accordance with the Supreme Court’s Order…provided this Court with the power to act.”

Also Wednesday, Vance’s office filed an application with the Supreme Court asking the justices to enter the judgment so that the lower court could swiftly address the new legal challenges. The court typically takes 25 days to do so after an opinion is issued.

The district attorney’s office urged the district court to proceed in a speedy fashion, writing that “this Court should conduct any remaining proceedings with the same expedition it dispatched when it first rejected the President’s allegations in fall 2019, in light of continuing concerns about the potential loss of critical evidence and expiration of statutes of limitations.”

Vance’s office said judges had already rejected some of Trump’s arguments, including that the subpoena was politically motivated. Vance’s office pointed out that US District Judge Victor Marrero months ago found “there was no demonstrated bad faith, harassment, or any other unusual circumstance that would call for equitable relief.”

The district attorney’s office emphasized that the Supreme Court had rejected the idea of a heightened standard for a state subpoena issued to a sitting president, saying Trump was attempting to “elide that standard.”

“Equally important,” lawyers for the office wrote of Trump’s proposal, “it overlooks the fact that he has already advanced substantially similar allegations in the Amended Complaint, which this Court rejected.”

Vance’s office again turned to Marrero’s earlier ruling to dispute the potential argument that compliance with the subpoena would be burdensome for Trump, saying the judge had previously rejected that idea.

Vance’s office described Trump’s suggestion of discovery “premature,” and wrote that even if the President were to file an amended complaint to spur discovery, “discovery into the district attorney’s motives would be highly irregular and inappropriate.”

The district attorney’s office said it would agree to not enforce the subpoena seeking eight years of Trump’s tax returns and financial records from his long-time accounting firm Mazars USA until July 27, or if Trump files a new lawsuit it won’t seek to enforce the subpoena until seven days after that court makes a decision.

Both sides also agreed to an expedited schedule, which if approved by the court, could mean the motions would be completed in the case by mid-August.

A hearing in the matter is scheduled before Marrero Thursday morning.

This story has been updated with additional information.