To some infinitesimal amount, possibly. It depends on how far the chamber is from the Earth rotation axis as compared to how far the muzzle was from the Earth rotation axis. I presume my post was fairly understandable and your question is just poking fun. But if not, let me know.
No, no “poking fun” intended. This is scientific discourse. If in the northern hemisphere shooting south, as you say, the target is moving east faster than the rifle. however, the rifle’s muzzle is moving east faster than its chamber. Therefore it seems that the barrel movement would offset any need for aim compensation.