There was Foldimate, a startup which got publicity in 2019. They went bankrupt.[1]
Willow Garage demonstrated folding with their mobile robot. They shut down when the funding ran out. There was Laundroid, priced around US$ 15000, bankrupt 2019.[2]
Chicago Dryer's machines can fold shirts and pants. This is mostly used for commercial workwear, such as hospital uniforms. They have all the steps in the process automated. If you buy all the options, you can start with a bin of damp laundry direct from an extractor and get folded items out. (Other companies make washing systems with automated load and unload to feed the "clean side" of the process.) There are vision systems, machine learning, and robotic grippers inside those big machines.
So home-sized is possible, but not cost-effective. Boring old Chicago Dryer is cost-effective compared to low-wage labor, and they have real products that sell.
[1] https://www.facebook.com/FoldMate/videos/who-likes-folding-l...