5vr hooked up to truck being towed down the road is all fine and well then why in the heck would it HURT anything staying hooked up while stationary.
No bouncing up/down, jerking back/forth, tilting side to side etc.
Fridge.if turning on for cool down.....reasonably level....good to go
Umbilical cord.....
SOME trucks the 'battery charge' line remains hot with truck ignition key off.
Ford isn't, Ram is...Chevy/GMC not sure
So if 5vr battery is LOW then truck battery could be drawn down.
Hooked up, umbilical cord still plugged in and on shore power..why not!