Yes, they get visually more sphere-like as more faces are added. But spheres are obviously/trivially non-Rupert, while the question of whether a convex polyhedron can be non-Rupert is more interesting.
Would be interesting to see how much sides you can keep adding before the shape can't pass through itself. Or maybe you can indefinely keep passing them through, occasionally encountering noperts. Or maybe the noperts gradually increase, eventually making the no-nopperts harder to find. Who knows, let's find out.
You'd probably end up with tighter and tighter tolerances such as they mention with the triakis tetrahedron.
The challenge is that it gets computationally intensive the more sides that you add if you don't have shortcuts like ruling out entire blocks of orientations in their parameter space (they figured out that if one shadow, projection, protrudes significantly, then you'd need a large rotation to get that protrusion into the other shadow, thus removing all of those rotational angles and reducing the number of orientations needed to check). More sides and more symmetry make it much harder to test a candidate, but you have an interesting idea.
