If it's sealed and no (appreciable) dirt has gotten inside, I suspect (as I do with my Helipot collection) that it's corrosion. I have doubts that Deoxit will work, but I've thought the same as you about poking a hole and filling it with a cleaning slurry, even though it probably won't work. I think the fundamental issue is that the "wiping action" is good for bits of dirt, but that corrosion is a much more tough customer.
I think for corrosion, some form of chemical attack might do it, but I also think that that would change the value a bit.
I have a perspective on this:
In parallel with whatever miracle cure you're contemplating, I would think about getting another pot, at least as an option. Since I don't have, and can't find, a schematic for the thing, I can't speak definitively to the calibration issue, but there's a fairly strong chance any tweaking you might have to do is still in the ballpark of the calibration of the thing as it sits. You have the advantage that error contributions from that digit(s) is a thousand times smaller than that of the first decade. The only thing that might matter is the total resistance, and if you can measure that with an accuracy of 1000 times worse than the 931b's specs, you can stay pretty close to "good" when you replace the pot. Tweak it with a big-old parallel resistor or whatever you like to get the total back where it's supposed to be; that will be part of the fun of it all.