The month on my Patek Philippe 5270G-001 I purchased last month from a large Patek AD (who also sells preowned, one of the largest in the world ) on April 3rd will not change over. The issue didn’t become apparent to me until May 1st, which was the first changing-of-the-month since I purchased it on April 3rd. On May 1st, it never changed over from April to May in the month window — the “APR” simply dropped a few millimeters but never actually changed over. At that point I became quite concerned. It gets hung up slightly and doesn’t change. All other windows, functions, etc. work and change over flawlessly. I tried all 12 months, set it to the 30th or the 31st allowed it to manually tick over and all other complications related to the date change over except the month which gets hung up and never changes consistently across all 12 months. What is the issue here and is there anything I can do? I am quite unhappy about this and having to send this watch off to Henri Stern not to be seen for 6 months is just not acceptable to me even if the cost of repair is taken care of. The Patek AD says they have other watchmakers who they work with that can fix this which is not at Henri Stern. When I questioned them, they told me they have worked on Sky Moon Tourbillon, etc. and this was an easy and quick fix and no need to send the watch to Henri Stern for it to disappear for six months. I’ll refrain from naming the AD for now, but this whole situation is very concerning. A simple refund isn’t possible as the deal involved a trade of two other watches of mine that have since been sold. What do you guys make of this? What recommendations do you have?
Here it is four days into May and still not changing over:

And here is a test from December 31st to January 1st — same issue: