Looks like it was declared unconstitutional in 1973.
In 1973, Federal District Judge William Copple ruled that the statutes in question violated the Constitution of the United States and that Arizona could not keep the party off the ballot (Blawis v. Bolin).
CPUSA doesn’t need to field any candidates because the Democrats are running the exact same platform.
Very few “liberals” would vote for hard-core communists, but voting for Dems doesn’t feel as icky.
The outcome is the same.