After a long couple of weeks, the phone is finally on its way back from the Samsung repair center.
According to the online service tracker, the LCD screen was replaced. When I sent it in, there wasn't anything wrong with the screen though. The screen was fine, it just didn't boot up.
I'm hoping that they at least powered the phone up to make sure it booted up all the way before shipping it back. I don't think I'll be very pleased if I get the phone back with a new screen but having the same problem still.
Should be arriving early next week, so I'll have to wait and see what I get back.