The general case might be something like that, but you only need any given configuration that fits those parameters (that is, you're not trying to solve complete, abstract topological problems). Maybe if you arbitrarily fix the first path, thereby artificially reducing the solution space? Just thinking out loud. Very cool puzzle!
EDIT: or even take a lower-level solution and inject another path into. So every prior level is a seed to future levels. Rotation of the three faces, even cycling the colors, would probably make it hard to recognize the older level within the new one (at least, for casual use).
EDIT: or even take a lower-level solution and inject another path into. So every prior level is a seed to future levels. Rotation of the three faces, even cycling the colors, would probably make it hard to recognize the older level within the new one (at least, for casual use).