Wouldn’t this problem be related to the problem of finding whether two shapes collide in 3d space? That would probably be one of the most studied problems in geometry as simulations and games must compute that as fast as possible for many shapes.
It's pretty easy to brute force most shapes to prove the property true. The challenge is proving that a shape does not have the Rupert property, or that it does when it's a very specific and tight fit. You can't test an infinite number of possibilities.