On May 8, U.S. Senator Robert Bennett lost his fight to get enough votes at the Republican state convention to be able to get his name on the June 22 primary. Bennett has been a U.S. Senator since 1992. See this story. Utah law says candidates who don’t get at least 35% at the state convention may not get their names on the primary ballot.
It is unknown if Senator Bennett would run as an independent if the law permitted him to do so. The story accurately explains that the deadline has passed; it was March 15. But if he did wish to run as an independent, he could probably overturn the deadline in court.
Many courts have ruled that independent petition deadlines, or even independent candidate deadlines for filing a declaration of candidacy, for non-presidential office, cannot be earlier than the primary (or the day before the primary). Such decisions have been won in Alabama, Alaska, Arkansas, Illinois, Indiana, Kansas, Kentucky, Maine, Maryland, Massachusetts, Missouri, New Jersey, Ohio, Pennsylvania, and South Carolina. The most important courts that have issued such rulings are the U.S. Courts of Appeals in the First, Third, Fourth, Seventh and Eleventh Circuits. However, Utah is in the Tenth Circuit, which has never had a case over the deadline for non-presidential independent candidates.
Also, in 1977, the U.S. Supreme Court summarily affirmed a decision of a 3-judge U.S. District Court in Arkansas, striking down an April petition deadline for an independent candidate for state office. The Arkansas primary at that time was in May.
The only circuit that has upheld such deadlines is the Fifth Circuit, which upheld Texas’ declaration of candidacy deadline for non-presidential independents in 1996. That deadline is the first week in January.