It appears to me that the petition filed was actually a combined petition for panel rehearing and a petition for en banc review. So, before you even get to the question whether there should be en banc review, there is the initial question whether the panel itself wants to change anything in the opinion. So, with respect to whether there is a call for a response, I think it will first be up to the panel, and I would imagine that one judge could call for a response, but if it's just the Chief and the other two don't want a response, who knows what happens.
I don't know Second Circuit practice at all, but in the 9th Circuit, what occasionally happens is you'll file a motion for panel rehearing and petition for rehearing en banc, and time will go by, and then one day you'll get an amended opinion from the Panel, where they fix one of the problems you've identified in an amended opinion (usually you still lose) or maybe they'll clarify what they say to harmonize the case with a case from another circuit or from a prior panel opinion. Inevitably when this happens, you then get an order that says the court having amended the opinion, no active judge voted for en banc and en banc is denied.
Even that is pretty rare -- usually you just get an order denying both. My main point here is that most -- including in the media -- have jumped straight to the en banc procedures but with respect to procedure and theoretical (but not likely) outcomes, the fact that there is also a petition for panel rehearing does matter. (A petition for en banc is virtually always coupled with a petition for panel rehearing, with the thought that the panel should be given the first opportunity to fix their alleged mistake, but not always.)
Edit: Looking at the second circuit's local rules, they are different from what I'm accustomed to in that if the panel grants rehearing and "substantively" changes its opinion, a new petition for rehearing en banc is permitted from the amended opinion. I think that may also be the rule in the 9th Circuit, but they typically shortcircuit it by simply denying en banc on their own after the amended opinion -- though I suppose procedurally there still may be a way to seek en banc again.