On January 16, U.S. District Court Judge John Tuchi, an Obama appointee, issued a ruling in No Labels Party v Fontes, 2:23cv-2172. It says that No Labels Party has a freedom of association right to bar anyone from running in its primary for congress or partisan state or county office.
In Arizona, all qualified parties nominate by primary. Arizona is in the Ninth Circuit, and the Ninth Circuit had ruled in an Alaska case that the Alaskan Independence Party could not prevent a particular person from running in its primary. But Judge Tuchi distinguished that case from the current No Labels case, by saying that in the Alaska case, the state had an interest in preventing a corrupt party leadership from determining which individuals could run, which is different from an instance when a party has a blanket ban on anyone running for certain types of office. Here is the ruling. Thanks to Richard Grayson for this news.