FretShackPatrick

Hey! This is Patrick from the Fret Shack.

I see that a couple of people have already responded, but I'll still put in my two cents.
Any point of contact is where friction can be introduced and cause binding, which in turn will affect tuning stability i.e the tuners, nut, bridge, and in the case of Biggby's, the roller for the tail piece.
While all these points of contact may cause issue my guess would be the Tune-O-Matic style bridge. Due to the downward angle of the string from the bridge to tail piece, it wants to bind in the saddle. sometimes from the factory, they don't cut the saddle slots well, leaving them with hard edges that the string can catch on. Recutting and polishing the saddle slots will help, but nothing will beat a roller bridge, which should be a vast improvement to tuning stability.
Another upgrade to do would be the front roller. There are a couple of different systems, but I've seen that the Callaham upgrade seems the easiest to obtain and install.

If your issue continues, I would take a look at the nut could just be that the nut isn't cut properly.