I didn’t even know there were different versions of Turner’s Cube. 
But, really, this was more in the line of idle speculation. Once I started thinking about the degree of stock prep work involved, my brain started chasing the rabbit, and went from cubes to “x-faced Platonic solids,” and kept thinking “there’s got to be a better way to do this, some trick I’m not seeing.” But, sounds like this is one of those things you have to do the hard way (or by making an optimized jig of some sort).
For the purposes of being useful in this forum, I would think we’d want to limit ourselves to 3-axis machines, like the Nomad. Although N-axis “war stories” are always welcome!
(No one took the bait on the “tesseract” joke? And here I was waiting for someone to make the “5-dimensional mill” joke…)