Connection lost for no reason at all.

My M650L mouse lost connectivity.
New battery. It just stopped moving in the middle of operation.
To my luck(?) I have another same model, and it functions as expected, on the same laptop.
I had a chat with the Logitech support, and they promised me, they would help, but when all their usual basic guides didn't help, they stopped responding.

Did anyone manage to fix the famous connectivity problem with this mouse?