The stalemate between Mr. Kraft’s high-priced legal team and law enforcement officials in Palm Beach County hinges on two things: The terms of a proposed deal and the status of any video evidence of Mr. Kraft.

Last week, prosecutors offered to drop the two misdemeanor charges against Mr. Kraft, but the proposed deal required him to admit that he would have been found guilty at trial on the charges. Many defendants in these types of cases routinely accept such terms to avoid the cost and potential embarrassment of fighting them.

But Mr. Kraft is not backing down. His legal team and prosecutors spent much of last week trying to get prosecutors to alter the terms of the deal offered to him and 24 other men who were similarly accused of soliciting prostitution at a massage parlor in Jupiter, Fla., called Orchids of Asia Day Spa. But Mr. Kraft and many of the other men are unwilling to admit guilt because they believe the police mishandled aspects of the case.

They contend that a search warrant was improperly obtained to install secret cameras inside the massage parlor. According to the police, the video shows Mr. Kraft and the other men soliciting sex. The pretext for that warrant was predicated on the existence of human trafficking at that massage parlor, something that the police have not yet proven and that Mr. Kraft’s lawyers insist did not exist.

At the same time, Mr. Kraft and 14 of the other men last week filed a motion asking the court to ensure that all evidence in the case, particularly the surveillance video, be kept private to avoid bias among potential jurors. That request prompted an unusual response from Sheriff William Snyder of Martin County. He said the video evidence would most likely be made public once the case ends.