On March 13, the Connecticut House Government Administration & Elections Committee introduced an omnibus election bill that includes a provision making it illegal for any qualified party to have “Independent” as part of its name. The bill, HB 5498, can be seen here. The provision about party names is on page 20. It also bans “unaffiliated”, or “unenrolled” in party names.
The Independent Party of Connecticut has been ballot-qualified for most federal and state offices starting in 2008. If the bill becomes law, it would be forced to change its name by January 1, 2025.
Here is a news story about the bill, which had a hearing on March 18. The party testified against that part of the bill. The committee hasn’t acted on the bill yet.
In 2019, California Governor Gavin Newsom vetoed a similar bill. Here is his veto message. He said the idea violates the First Amendment.