I also have the same issue, I tried to resolder the battery pack connection but it did not solve the issue. If you tried the cap then I don't think by changing it for me will do any good.
Before this issue it was the fret buttons. I fixed them by cleaning them and fixed the neck issue by using a bit of cardboard. I played a few week and this issue started to appear when I bought a second guitar controller to play co-op. I thought it was the xbox that have connection issue with two controllers but the new one works flawlessly. So I tried the old one on single player but it has the same symptom as you described. Still cannot figure where on the board is causing this random disconnections.