I don't believe they even have to originate their public service block from said "main studio." They just have to have a working EAS, the public file, and the ability to broadcast from the location. The FCC defines what is needed to be considered "able to broadcast." I remember it includes mixing equipment, but I'm not sure what else. The "main studio" is also required to have a management level employee there during business hours. While said manager doesn't have to be chained to his desk, he is expected to keep reasonable and appropriate business hours.
A classic case of this would be KBFL-FM 99.9 in Buffalo, MO. All the programming originates from the KTXR studios in Springfield, well outside of the station's citygrade signal contour. However, it very much has a main studio in Buffalo. It's actually a pretty nice building, and there is indeed a manager there during regular business hours. When leaving for lunch, they put a sign on the door saying they're temporarily out of the office and will be back later.