...and it depends on the truck, too.
On my Dodge, the +12v charge line is always hot, whether the truck is running or off. This feeds the 12v
circuits in the trailer, so everything should work.
I've done this purposefully to get a "boost" toward the end of a boondocking outing, when the trailer battery is getting weak, lights are dim....plug in to the truck, and "poof"--lights brighten up.
Just know, you're draining your tow vehicles battery when you do this.
Other makes/models don't leave the +12v hot, unless the vehicle is running.