Cool picture - thanks for posting.
I wonder if there are louvers or dampers on each of the incoming vents to prevent back-flow into the other non-running vents. You're right about the constant suction, Toggle, that would help the system tremendously - especially if the exhaust line from the box has to travel some distance to reach the outdoors.
An architect friend of ours added a garage onto the front of their house that interfered with the dryer venting to the outside. Their Speed Queen dryer caused an objectionable lint problem on the outside so allowing it to run into the garage was out of the question. To solve both problems, he built an enclosed box around the vent and added a louvered eave vent to the outside that had a screen on the inside to prevent bugs, etc. from entering. Every couple of months, they would open the top of the box, and brush or vacuum off the screen and accumulated lint in the box. I personally liked the lint residue on the outside of the house - sort of a badge of a clean family ;-)