Good investigation, if you have determined that the shaft that holds the sprocket is the issue.
Now if your diagnosis is correct, you need to have this camera repaired. Either the bearing for the shaft is loose or the bearing for the gear attached to the shaft is loose, and it will ultimately jam solid.

I have had two M bodies do this, and one SL body.

You can't fix it from outside the body, and lubrication won't help. The top cover needs to come off, the gear isolated and the bearing tightened down.

If there isn't anyone in Denmark to do this, then send it to Leica Solms.