Thank you your post was most helpful as Just received a used Line 6 JTV-59 Variax and after pressing the Knob the guitar selection knob wouldn't turn on. I could update firmware and use Workbench. It was frustrating because the software could see the JTV-59. After reading your post I removed the knob and pressed down and selection light came on and everything worked.
Obviously it had not been checked out from the retailer. The knob does seem to have a lot of play side to side and the tuning knob does not so will need to inquire about that.
The wad of paper works just fine and I now have a working guitar.