Unfortunately, I imagine that the answer will be tangled up in whether or not Andy Karl gets a Tony nomination. If he does, I'm sure that both he himself and the producers would like him to continue with the show (even if that means modifying the staging) through the Tonys in hopes of winning. I don't know how long he'd be out of the show for surgery, but I would imagine it would be several weeks, if not a couple of months. Unfortunately, this is exactly the wrong time for all involved to do that.
If Karl is nominated, it wouldn't surprise me a bit to see him limp through as many performances a week as he can manage and to go out of the show for surgery shortly after the Tonys.