That's a good one.
First, I'd bypass the switch to see if that's the problem.
Then I'd do a voltage reading under load at the middle fixture to see if there is a bad connection somewhere causing a VD or intermittent power.
Then, I'd go through the lamp sockets and internal wiring of the fixtures to see if there is anything causing the short-circuit protection of the ballasts to activate. Get the fixture lit and then jiggle the lamp a few times to see if causes the fault/shutdown.
Good Luck!
