The latest news on this front comes from a really interesting article at Ballot Access News written by the widely-respected election expert and advocate for third party/independent ballot access Richard Winger. Take a look at this excerpt:
As already reported, on March 29, 2021, two important ballot access cases were won in federal court, in Michigan and Georgia. In Illinois and Maine, there are pending ballot access cases in U.S. District Court; and in Arkansas, there is one in the U.S. Court of Appeals. Attorneys for the plaintiffs in Arkansas, Illinois, and Maine promptly informed the judges in those states about the Georgia and Michigan decisions. So far, attorneys for state governments in Arkansas, Illinois and Maine have not filed their own briefs commenting on these new precedents, even though court rules permit them to do so.
The Arkansas, Illinois, and Maine decisions could come out at any time. The Illinois case, filed in 2016, challenges the 5% petitions for independent candidates for U.S. House. The Maine case, filed in 2019 by the Libertarian Party, challenges several ballot access laws relating to new or small parties. The Arkansas case, filed in 2019, challenges the procedures for non-presidential independent candidates.