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I reported this to the most visible Square One guy I could find on the web (Jaap Scherphius), and he sent me here. Yes, I have spent the last year or so computing God's Algorithm for Square One, in the center-flip-count domain, taking advantage of all top/bottom rotational degeneracies available. The table calculation itself is finished, but I am not finished analyzing it yet. I can report that the maximum number of flips required to get back to START is 13, with many shapes (not just square-square) contributing to that number.

The main stumbling block to reporting results at the moment is that I discovered a unexpected position-counting problem, where the shape degeneracy interferes with the 16-fold starting position degeneracy, and which requires me to do a recount on the tables. I will post the results back here in a few days when the recount is finished.