They surely did.
First of all there was the switch from old post spacing to new and even though I never measured a set I think the difference in post spacing is enough to assume the plates were not same. From studying photos and direct measurements I know that there were at least five different versions just on Loar signed F-5s (mostly noticed by three or five mounting holes) starting with wiggle end plates on early models (like F-4 tuners but with pearl knobs), then the long spacing arrow heads (some of them worm above?), then short spacing arrowheads followed by the three hole version and after Loar era they went to worm above... etc, the tuners changed all the time. From manufacturing point the mounting plates were made out of longer strip and cut to length using dies so slight differences even within batch and especially during the transition times can be expected (mixing leftover old plates with new but punching the holes for posts later).
One thing that folks should remember on these tuners worms are assembled first (riveted) and then plated which adds material to the gears and may cause harder turning (if they didn't compensate for that) till the plating gets worn off the contact surfces. Dry turning with drill won't help, you need to add some kind of grinding paste that will cut material away. Basicly that's the old school way of grinding worm gears. The cutting head is similar worm with teeth and it cuts and at the same time rotates the wheel till the final depth of cut is reached. I cannot see reason why this wouldn't loosen ANY tuners (unless relly visibly screwed during manufacture).
Several times I solved problems with tuners it was the stepped holes that caused problems. Many makers drill holes for bushings only halfway through headstock and the rest of hole is 1/4" or so which is pretty much exact size of post on many tuners. The taper of headstock creates need for slight angle of posts through wood when bushings are mounted square to face of headstock but tuners square to rear making it easy for posts to jam in place. Once dirt accumulates or even dry weather will make the holes too tight and tuners won't work well. From perhaps hundred or so mandolins with problematic tuning I only had to change tuners once when the originals were worm under reworkde to worm above by drilling off the riveted tabs and soldering them back in reverse order - and with relly bad misalignment - they've worked for few decades even in that crude state.
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