You can find the cert. petition in Utah Republican Party v. Cox at this link. Here are the questions presented:

As a private expressive association, “[a] political party” enjoys a general First Amendment right “to choose a candidate-selection process that will in its view produce the nominee who best represents its political platform.” N.Y. Bd. Of Elections v. Lopez Torres, 552 U.S. 196, 202 (2008). The First Amendment thus gives “special protection” to “the process by which a political party selects a standard bearer” California Democratic Party v. Jones, 530 U.S. 567, 575 (2000). Here, however, the Tenth Circuit has joined the Ninth in permitting a government to force a political party to select candidates through a primary rather than a caucus system, for the viewpoint-based purpose of avoiding candidates with “extreme views.”

The questions presented are:

1 Does the First Amendment permit a government to compel a political party to use a state-preferred process for selecting a party’s standard-bearers for a general election, not to prevent discrimination or unfairness, but to alter the predicted viewpoints of those standard-bearers?

2. When evaluating the First Amendment burden of a law affecting expressive associations, may a court consider only the impact on the association’s members, instead of analyzing the burden on the association itself, as defined by its own organizational structure?

The brief cites to my earlier writing on this question:

Richard L. Hasen, “Too Plain for Argument?” The Uncertain Congressional Power to Require Parties to Choose Presidential Nominees Through Direct and Equal Primaries , 102 Nw. U. L. Rev. 2009, 2010 (2008)