I bought it from a guy on the analog list and the thing is
cosmetically almost perfect. It clearly sat in a studio and wasn't
used much. The previous owner warned me a particular problem and
after it warmed up, I've seen it as well.

After I played with it for about 1/2 an hour, I pressed the "tune all"
button. Afterwards, certain notes were out of tune with the others.

I pressed the tune one button, but it doesn't fix it.

Any ideas?

I think the synthesizer otherwise is fine. I think. Hard to tell.

But all the buttons, keys and membranes are working, so that strikes
me as a good sign.