One rail of my three rail track lighting fixture isn't working
You probably have 4 rails there, just two happen to be in-line, on either side of the power input (gray block in center.)

Your problem is likely at the right side (in the picture) corner connector. Given that the power supply appears to be shared, the power supply would seem to be fine given the working lights - but the conductors are not conducting around the corner. Exactly why is hard to determine remotely, but that's where to look.