ANCHORAGE, Alaska — An Anchorage Superior Court judge on Tuesday agreed to expedite a lawsuit whose outcome will determine whether the namesake challenger of U.S. Sen. Dan Sullivan will appear on the state’s Aug. 18 primary ballot.

Superior Court Judge Thomas Matthews, in a status hearing Tuesday, said the case was moving at a “very, very fast pace.”

“I can make every effort that I can to get a decision out as quickly as possible,” he said.

The challenger, Dan J. Sullivan of Petersburg, filed an appeal to the court Monday after Carol Beecher, head of the Alaska Division of Elections and a Republican, disqualified him from the ballot last week.

Beecher had argued that he had not filed to run in good faith, and alleged that he instead wanted to confuse voters. She has said that Alaska administrative code prohibits her from placing a name on the primary election ballot “in a manner that is confusing or misleading to voters or compromises the fairness or neutrality of the ballot.”

Now, the courts are racing to consider the case, to beat an upcoming ballot-printing deadline.

Matthews, in a preliminary hearing Tuesday, said he would set oral arguments for Thursday at 10 a.m.

The hearing, with 30 minutes for each side, is expected to be livestreamed.

That will give the media a chance to listen in, he said.

“There’s great public interest,” Matthews said. “Obviously, we want to get it right, and get you all a decision just as quickly as possible.”

The case raises constitutional questions in a contest that could help determine control of the U.S. Senate after the November midterm elections. Previous court rulings have determined that the bar for removing a candidate from Alaska’s ballot is exceedingly high.

Beecher’s decision to keep Dan J. Sullivan, who registered as a Republican, off the ballot sided with the Alaska Republican Party and the National Republican Senatorial Committee, which claimed that the challenger was attempting to thwart the reelection bid of the Republican incumbent, Sen. Dan S. Sullivan.

The groups have suggested that Democrats encouraged the Petersburg Sullivan to run, in order to benefit the incumbent senator’s top opponent, Democrat and former U.S. Rep. Mary Peltola.

Peltola’s campaign has denied any involvement in the issue.

Dan J. Sullivan, 70 and a retired teacher, has said he meets every qualification to run. His attorneys in the case argue that Beecher has violated state law and the U.S. Constitution. They assert that Beecher’s decision has “disturbing implications” that could allow her to disqualify candidates “based on nothing more than her own view of who voters should be allowed to vote for.”

Dan J. Sullivan is asking Matthews to overturn Beecher’s decision, allowing him to appear on the ballot.

If the courts agree to that step, the Division of Elections could eliminate voter confusion by “printing Mr. Sullivan and Senator Sullivan’s full names on the ballots, with their respective campaigns of course free to promulgate whatever messaging they feel is appropriate to alert voters of the presence of and differences between the two Sullivans on the ballot,” attorneys for Dan J. Sullivan argued in a filing.

The judge could also take a different path, granting a preliminary injunction to stop the state from printing ballots until the issues in the case are resolved, the filing says.

An injunction would “allow the election to proceed in an orderly manner if the Court ensured that the parties were required to comply with an appropriately tight briefing schedule,” the filing says. “A risk of disrupting the larger election timeline is simply not present.”

The Alaska Republican Party is seeking to intervene in the case.

Matthews said Tuesday that he’ll address intervention proposals “as quickly as possible.”

Beecher has previously said the state’s ballot-printing deadline is Sunday.

But Rachel Witty, an attorney for the state at the hearing, provided the judge with a new “drop-dead” printing date, Tuesday at noon.

Witty said the goal would be to have a decision by Matthews on Thursday, so the case can be moved to the Alaska Supreme Court on Friday.

That would give the Supreme Court the opportunity to make a final call by next week’s deadline.

Witty also told the courtroom that she has notified the state Supreme Court that the case might soon be on their desk.

“I think they’ve got their eyes peeled for this,” she said.

Dan J. Sullivan did not immediately respond to a text seeking comment Tuesday afternoon.

Jeffrey Robinson, an attorney for Dan J. Sullivan, declined to say after the hearing who is paying for the challenger’s legal representation.

“I’m not making any comments on ongoing litigation,” he said. “I appreciate your question.”

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