Perhaps a random thought, but the bitcommitment primitive in bitvm seems to extend naturally from 2 to N different outcomes. Equivocation means 2 out of N preimages revealed, which for smallish N is ok, but it's combinatorial.

I can't help wondering if you could get fancy to make huge N values work: do a Lagrange interpolation/Shamir setup for 2 of N, such that seeing any 2 reveals the secret. Would need to be a 'point lock' not a 'hash lock', of course.