Looking at the Psychic gym, if you pause at the last possible moment, you can see two blue panels on the ground. I would assume that these rearrange the room to allow you to progress through the gym when the switches are stepped on.

Remember those two pillars of fire? If you pause before they erupt, you can see several patterns on the ground that indicate multiple doors, possibly four. This leads me to believe that this might in fact be the Elite Four, and the pillars indicate that the champion is now accessible through the central door.
Or it could just be a late game gym or something.