A video of you lunging would be really helpful here, just to be able to more accurately interpret your description of what’s failing.

If you find you collapse TOWARDS the right hip, it may be a matter of your lead left leg losing stability and integrity. Does your arch collapse and your knee dive in as you descend? If so, try screwing your left foot into the floor (knee towards your pinkie toe) as you perform the movement.

You would probably benefit from spending some time working on static stability in the bottom position of the lunge. Kneel down on a pad, set yourself up in that idea lunge position (front shin vertical, butt tight on the back leg side, foot behind your knee), and hold that position for sets of 30sec (or however long you can manage). Then, try training your split squat from the bottom-up.