Or the political leanings of the Judges on that panel?
Right off the bat, you can guess how 3 of the 10 judges will vote, because the three judges on the original panel are likely to vote the same way.
So, Posner and Flaum will likely vote to deny rehearing and Williams will vote to grant rehearing.
The most dependable pro-gun vote on the court IMHO is Sykes, so she is very likely a "deny" vote as well. She authored the Ezell opinion (striking down Chicago's shooting range ban) and was the sole dissenting vote in Skoien (unsuccessful challenge to federal gun ban for persons convicted of misdemeanors involving domestic violence). The for sure "grant" vote is Diane Wood, a liberal leaning member of the court nominated by Clinton. You can probably assume that David Hamilton (nominated by Obama) will also vote to grant the petition. Kanne (a Reagan nominee) will probably vote to deny the petition. Kanne concurred with Sykes in the Ezell opinion.
The remaining judges aren't so clear. Rovner (a Bush I nominee) did concur with the Ezell judgment, but authored a separate opinion that was less pro-gun. I think he will vote "deny" but I'm not sure. Easterbrook (Reagan nominee) is known as a conservative judge, but I don't think he is pro-gun (and may in fact be personally hostile to gun rights), but even if he doesn't favor gun rights I think, unlike liberal judges, he will follow the law honestly. He may vote to grant the petition for rehearing, but I think that would only be because he wants to make the scope of the opinion narrower even if he agrees that the ban is unconstitutional. I don't know enough about Tinder (Bush II nominee) to predict his vote, but at least he is a Republican-nominated judge.
So the total tally is:
Posner -- Deny
Flaum -- Deny
Sykes -- Deny
Kanne -- Deny
Easterbrook -- Maybe
Tinder -- Maybe
Rovner -- Maybe
Wood -- Grant
Williams -- Grant
Hamilton -- Grant
In order to grant Illinois' petition, there needs to be 6 votes. That means that besides the four likely deny votes, we only need one of the "maybes" to come to our side. Because all of the maybe's are Republican appointed judges, we have a good chance to get that last "deny" vote.