I'm sure that the physical part of this problem is also hard. But I have a lot more faith in robotics people coming up with the right "hands" with appropriate sensors on them to grasp a single piece of clothing and separate it from a pile, and then to manipulate that item on a large work area until it is folded. Maybe some delicate load sensors in the fingertips to adjust gripping force appropriately between the silk blouse and the corduroy trousers. Maybe no fingers at all, and just vacuum-and-pneumatic fabric handling devices. Or some other combination.

The things I've seen on assembly lines are beautiful and clever. But they all rely on a sort of consistency of input.

So yeah, that part is hard. But I think the intelligent control of whatever apparatus is used is harder still. To be able to recognize the different items, know when to turn them inside out, know when to bring the unbuttoned halves of a dress shirt together, etc. That's all very hard! Going from a chaotic pile of mixed clothing to a neat stack of folded garments is something a child can do easily, but no AI controlled robot in the world can do at all.

And if it fails on fitted bottom sheets, I won't dock it any points. Even I can't do that!

They're talking about the motion control and planning, which is certainly in the scope of what ML/AI is solving nowadays.

I don't think robotics are the part holding back a clothes folding robot.

If you hooked up some basic manipulators to a VR controller, I'm pretty sure I could fold some shirts.

Not particularly well or quickly, but to a level that I'd be happy to see from a first generation shirt folding robot.