Recussals by Sotomayor and Kagan (and no votes by Bader-Ginsberg and Breyer) are irrelevant to moving this appeal to a hearing before the full court.
It takes FOUR votes (the rule of four) to grant a Petition for a Writ of Certiorari, which is Supreme Courtese for “yes, we’ll hear your case.” There are five Justices appointed by Republican Presidents: Alito, Kennedy, Roberts, Scalia, and Thomas.
Sotomayor and Kagan are not needed.
If the rumor is correct with regards to the number of members who want to address this case, I am going to assume it is Alito, Scalia, and Thomas who are the ones who want to resolve the issue. Roberts would have to admit he suspected a problem when he swore Obama in and did nothing to address it. Kennedy is most likely sitting on the fence.