Sperzel Locking Tuners issue

My Les Paul Studio does not stay in tune quite as well as my Tele that I constructed. (In all fairness, this is pretty normal, because my tele stays in tune better than any other guitar I've ever played, including the Jimmy Vaughan strat I used to play, which stayed in tune damn well.) My tele has locking Kluson vintage style tuners, the ones you need a penny to unlock sometimes. Fantastic.

But my LP I bought used from a guy, and it has Sperzel locking tuners, the ones with a thumbscrew on the back of the headstock. Which means it should be pretty dead on too, I think. I took off the low E tuner because that tuner feels easier to turn than the others, and I wanted to see how they had been installed. Sure enough, the screw hole that the Sperzels require for their single spike to go into the headstock is too close to the old holes left by the Gibson tuners, making a larger double hole. It allows just the slightest wabble if you try to turn the tuner around in its hole without the front side screws holding it firmly in place. But if you screw it down firmly, it doesn't really SEEM to move around...? So...

1. I'm not even sure if this is an issue. Maybe the Gibson just isn't made as stably as my partscaster. IDK?
2. If it helps, the LP guitar holds pitch relatively well over a playing session, but the sustain of its notes is not as dead on as the tele. A note will waver, going a little sharp, then a little flat, according to all of my tuners, more on the LP than the Tele. The tele, you tune it to E, it stays on E, doesn't waver.
3. If I do decide to try to do something about it, what am I looking at? The baking soda and CA glue method? How do I do that? How hard is it to do?

Thanks. Opinions appreciated.

It doesn't sound like the tuner is an issue at all... sounds like the guitar itself, or possibly the nut. Seems like most guitar tuning issues originate from the nut.
I have those tuners, and I think they are good. ( My peg stuck in the wood when I took it off the other day pulled right out of the tuner! )
I might put some mixed wood glue and wood dust in the hole and re- drill it when it is hard as rock later.
Others will give better advice i'm sure.

Hmmmm... Yeah. The nut doesn't look like it has problems. And I use some Graph-it-all when I change strings.

.....?

Both my guitars have Sperzel, in my case when something sounds funny once I start to practice I've always found the cause to be the nut holding the guitar :razz:

You could try filling the hole with some 'bog', the stuff you use to patch up rust in your car, and re-drill the hole.

I've got Sperzels on a couple of my ASAT's, they're dead good and I've never had a problem with them.

Every Gibson I've owned, and those owned by friends, have gone out of tune far, far, more often than you'd expect from a guitar with a fixed bridge. Is one of the saddles loose at all?

I suggest you fill the double drilled area with dowel and Titebond. You may be able to fill the old Gibson hole and redrill for the peg.

I suspect the tuner locater pin is running out into the adjacent drilled area. If you overtighten you'll mar the headstock. Bondo or some polymerized filler will take a while to harden; I'd avoid that.