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:

At the oral argument on September 19 in Trump v Padilla, California Deputy Attorney General Peter H. Chang told the court, “Individuals don’t have the right to run for office.” However, the Copenhagen Meeting Document, an international human rights declaration signed by the United States, says “The participating states will respect the right of citizens to seek political or public office, individually or as representatives of political parties or organizations.” See 7.5 of the Document of the Copenhagen Meeting of the Conference on the Human Dimension of the Commission on Security and Cooperation in Europe, a document that the U.S. signed in June 1990.

That document is commonly called “The Helsinki Accords.”

See this article from Politico, which quotes Chang.