On August 6, legislative leaders from the New Mexico Democratic Party said they favor a bill in 2015 telling political parties that nominate by primary that they must let independents vote in those primaries. On August 7, the Republican Governor, Susana Martinez, said she also favors this idea. See this story. Given the Governor’s support, assuming she is re-elected this year, it seems very likely such a bill will pass.
These various public officials seem unaware that each of their parties already has the power to let independents vote in their primaries. In 1986 the U.S. Supreme Court said in Tashjian v Republican Party of Connecticut that if a party wants to invite independents to vote in its primary, that is a decision for the party, not the government. Nevertheless, it is obviously more orderly to have an election law statute setting forth how a party makes this decision, and also setting forth deadlines to make such a decision.