The mystery box is likely the timer delay. If you visit this thread:
You will find that a kind and resourceful fellow list member has added some electrical drawings for various gulfstream models. Read through the thread until you find a “GitHub” link, click it, and then look for drawing CE1461. You will see how the solenoid is expected to work. Your wiring might be a little different, but the principle will still apply.
Here is a link to the instructions for the Intellitec timer that Gulfstream typically uses:
As an additional thought, I found myself with dead coach batteries, brand new truck batteries, and extended slides. I pressed that “boost” button real hard, but nothing happened, and I had to pump the slides in by hand. The “boost” button will join the two electrical systems, but requires 12volts from the **coach** batteries. You might think about adding a second boost switch that is powered by the truck batteries so that you can join the two systems regardless of which system is dead. It’s a good while-you’re-in-there project.