Yes the cat manages, but not in this situation.
It might have been interesting to re-run the test with a more obvious "horizon" and see if the cats react to that. Then if the cat still doesn't right itself, it seems like it is using acceleration rather than vision to determine which way is down.
This is not possible because objects (including cats) in freefall do not experience the sensation of acceleration.
I'm fairly sure you just read this backwards, thought I'd explain better.
This is interesting because, it provides information on how the cat brain might function - if they have accelerometers embedded in their brains, that is quite interesting, but somewhat unlikely (birds have magnetic sensors, not impossible or unprecedented), if, perhaps more likely, they have incredibly sensitive hearing or sensation on their fur, which they interpret as acceleration, that is also interesting, in a different way.
But in a plane, you don't have the air rushing past your whiskers.
Edit: Probably that submission led to this one.