Washington \u2014 While President Trump fights in court over his remaking of the John F. Kennedy Memorial Center for the Performing Arts and construction of the White House ballroom, plans for the two Washington institutions are moving ahead.Now, the latest court filings in those cases are providing a window into the scope of work taking place at the White House on the site of the now-demolished East Room and Mr. Trump\u0027s visions for the Kennedy Center.At the White House, the ballroom project involves millions of pounds of steel and rebar, miles of conduit and a crew of more than 200 working nearly around the clock. And at the Kennedy Center, its board of trustees has advanced a plan to put Mr. Trump\u0027s name back on the building following its court-ordered removal and to honor him in at least one other way.The Kennedy CenterA status report filed\u00a0Wednesday with the U.S. district court in Washington provides a peek into changes to the performing arts center that its board of trustees agreed last week to undertake.The board is stacked with Mr. Trump\u0027s allies and administration officials, and chaired by the president himself. It voted last week to add Mr. Trump\u0027s name to the Kennedy Center\u0027s facade, a move that came two months after U.S. District Judge Chris Cooper ruled the board acted unlawfully when it put the president\u0027s name\u00a0on the building last year.On the heels of that decision, construction crews removed the president\u0027s name from the facade overnight on June 13. That area of the center\u0027s front portico has been covered by a tarp since then, though, concealing most of its name.But the board has again taken steps to put Mr. Trump\u0027s name back on the front of the Kennedy Center. A\u00a0resolution approved by the board last week calls for the inscription on the front of the building to read: "The John F. Kennedy Memorial Center for the Performing Arts Restored and Renovated by President Donald J. Trump."\u00a0It also requires recognition of the "president\u0027s contribution" to the center when the Trump Kennedy Center Fund reaches $100 million. Once that happens, the resolution states that the inscription on the building will read: "The John F. Kennedy Memorial Center for the Performing Arts Restored and Renovated by President Donald J. Trump Endowed by the Trump Kennedy Center Fund."The board-approved measure also requires the "physical site and grounds" of the Kennedy Center to be named "President Donald J. Trump Plaza."The board has said it will not implement these changes until Sept. 8 at the earliest.In the filing, submitted jointly by the Trump administration and lawyers for Rep. Joyce Beatty, an Ohio Democrat who sued last year over the Kennedy Center overhaul, both sides also addressed the tarp in the front of the building.The Justice Department said the "protective covering" and scaffolding were erected because of water testing and structural repairs to a roof overhang. One portion of that scaffolding was used to remove Mr. Trump\u0027s name from the building, government lawyers said.But they declined to say how long the tarp and scaffolding will remain in place, writing instead that its removal "is contingent upon the scope of any repairs required at present to the marble facade, as well as any other necessary repairs."They also indicated that the marble may have been damaged.During removal of Mr. Trump\u0027s name, the tarp "served as a safety barrier for ongoing construction activities and helped safeguard the marble facade of the building in light of reported panel damage," Justice Department lawyers wrote.Beatty\u0027s legal team said in the filing that the congresswoman is concerned that the Trump administration may have damaged the marble that lines the center\u0027s facade when it affixed Mr. Trump\u0027s name to it.White House ballroomThe White House ballroom has been a fixation of Mr. Trump\u0027s for months. But a federal judge considering a legal challenge to the project, brought by the National Trust for Historic Preservation, issued a preliminary injunction in March that would halt construction of the ballroom. U.S. District Judge Richard Leon has allowed below-ground work on national-security facilities to continue.Work above and below-ground has continued, though, as a result of administrative stays \u2014 temporary orders \u2014 put in place by the district court and a federal appeals court in Washington.But the Justice Department has now asked the Supreme Court to intervene and freeze the district court\u0027s injunction after the U.S. Court of Appeals for the District of Columbia Circuit ruled earlier this month that the White House cannot construct the ballroom without authorization from Congress.If the high court does not act by Friday, the D.C. Circuit\u0027s decision will take effect and construction of the ballroom must be halted.As part of the Trump administration\u0027s request for emergency relief, it submitted a declaration from Joshua Fisher, the director for White House Management and Administration who is overseeing the project, that sheds light on the volume of material being used for the project and the amount of work done already.The ballroom is being funded by private donors, and so far, $355 million of $400 million has been raised, according to Fisher. He said a 250-person crew is working 20 hours per day, seven days a week, and 65% of the project is done. Already installed are all below-grade structures, a grand staircase structure, a kitchen structure and a Colonnade movie theater structure, according to the declaration.The building will be made of concrete and steel, and Fisher said it will be completed by November. The facade is set to be "substantially" finished by April, and the entire project is expected to be wrapped by August 2028, according to Fisher. By then, Mr. Trump will have five months left in his term.The superstructure has been built up to roof-level, rising up to 70 feet...