I think you have a N.O. solenoid that is closing after power is shut off.
If this happens the panel will light up again.
The N.O. relay is the one that has 2 white/red wires attached. Next time it happens, if he unplugs the red 12v+ supply wire, the relay will be open. As in no power will be going through it.
It is also a cheap and common part to replace as a test. Any auto parts store should have them in the 40 amp 5 pin version. We also have these 28RA relays in stock.
I have a similar problem: New installation of a Beta 25. Battery connected as shown by the yellow + / - stickers on the engine. Engine cranks. (I don't have exhaust connected yet so I did nothing more than crank.)
Problem: Panel is live even with key "off" or even removed.
Where is the relay you spoke of? On the engine? In the panel?