Some people have installed the doors onto walls, and I think that would be your best approach. Here's one example:
IKEA Hack: PAX doors as room dividers and closet hiders » IKEA FANS | THE IKEA Fan Community
You'd have to cut the lip off the back of the track, and then install a piece of lumber or particleboard at the top of all the Paxes for the track to rest against and be mounted to. You'd probably want to make sure that is reinforced with L-brackets or something sturdy.
This would drop the doors down 1/4", but the brackets at the bottom track should have enough adjustment room to make up for that.
No guarantees it would work, but that's what I'd try.