If the whole neck (nut to neck pickup) is setup correctly by an experienced and competent luthier THEY WILL STAY IN TUNE.
I've done it more than once for my picky clients. Locking tuners also allow a proper setting of action at the nut (it always a compromise with a Floyd locking nut), plus string changes are a breeze compared to using a locking nut. ALSO - no weakening of the peghead.
Dave, I love ya - but you are DEAD WRONG on this one. Be wise and admit defeat.
My thoughts are based on experience. I've never played a Floyded guitar without a locking nut that stays in tune as well. Maybe a Golden Luthier can set it up so it does, but I'd bet most people don't have access to such a person. So there remains lots of guitars without locking nuts that don't stay in tune after a dive bomb.