Should be battery voltage. When you get 10.5V is the batteries showing the same? If so your batteries are weak, if they show 12+ volts and only 10.5@ plug then sounds like you are getting voltage drop due to a bad connection.
Will take a couple of GP cycles to narrow it down. Here is what I'd do. Have someone cycle the GPs while you monitor battery voltage. If it is higher than 10V then check voltage at the battery side of the GP controller, then the GP side of the controller. You could see a drop due to bad contacts in the controller/relay.
Let us know what you find.