With regards to improvement in IQ: this, while being (admittedly) a rather dubious claim, again is not necessarily impossible. Assume a set of optimally designed optics that will do what I described above without reducing IQ by simple introduction of more glass & air.
Then, what would happen is that when compared to using that lens with a glass-less adapter on the smaller (APS-C) format (this is the important point), all the aberrations etc. of that lens will ALSO be compressed, so that when the image is printed at the same paper size (IMPORTANT: NOT AT THE SAME MAGNIFICATION), then theoretically, you will see much less of those aberrations.
Again, this isn't anything magical. Think of the purple fringing you see around, say, tree branches against a cloudy sky background. The purple fringing is going to be less significant when you compare an image taken with that lens on FF (it's native magnification) than an image taken with that lens on APS-C (with 1.5x apparent focal length, which magnifies everything including purple fringing).
If, on the other hand, you bring the two images to the same magnification, then, you should have the same amount of aberrations, etc. You would expect exactly this, because we assumed in the beginning that the optics introduces no IQ degradation. In reality, it probably will, so when the images are shown at the same magnification, then there's probably going to be more aberrations with this product.
So you see, to repeat, there's really nothing magical or unbelievable or snake-oil-like thing being proposed here at all. Indeed the concept is simple enough that even I came up with it independently a few years ago while chatting with my brother.