According to this story, a bill will probably be introduced on Friday, March 25, setting up a system in which voters register into political parties. The legislators who are drafting this bill probably are unaware that several courts have ruled that voters must be permitted to register into an unqualified party. The story mentions that voters will be allowed to register into the Libertarian and Constitution Parties (which are both ballot-qualified). This implies that voters won’t be able to register into unqualified parties, such as the Green Party.
The Second Circuit and the Tenth Circuit have both ruled that voters must be allowed to register into active unqualified parties. Similar decisions have been made by a U.S. District Court in Iowa, and by New Jersey state courts.
That would be an advantage to just letting each party keep its own list of permitted voters (if it chose to). If the Green Party wanted to keep a list of voters it could. There is no reason that the list of permitted voters for each party be exclusive.