They'll polish it, but if it still visible after the polish they said it was my problem. They really discouraged polishing generally as well.
I have a lead for a good independent who will fix it. It will be all serviced and have fresh strap and likely look pretty good after that. Of course then Patek will wash their hands of the integrity of the case forever since it's no longer fixed to their specification. Will they even service it after that?
It's fine and in the scheme of things not a big deal, but it's just not "fun" and it's supposed to be fun. Like if I ding my proverbial Kia it can be fixed by Kia. They might use a body shop but they are authorized.