Wondering if anyone can please help.

I have a very old Airking that belonged to my grandfather. It's in pretty bad condition and I want it servicing by Rolex here in the UK. I'm not interested in using independents and will keep the original dial and hands (have done this before for my father's 6694).

The issue with the Airking is that the area between the lugs is heavily pitted and also worn, so the serial no. has completely worn away. Even using a loupe you can't see any numbering.

Will Rolex service this watch without a visible serial number? I still have the original box (he kept cigarettes in it!) but no papers.

Any help would be appreciated.