Finiteness conjecture for the sets J of multiple harmonic sums

Let $d$ be a positive integer, let ${\vec{s}}\in{\mathbb N}^d$, and let $J({\vec{s}}|p)$ denote the associated set of indices for the multiple harmonic sums. For a prime $p$, the set $J({\vec{s}}|p)$ is defined by the divisibility condition studied in the paper.

Finiteness conjecture. The set $J({\vec{s}}|p)$ is finite for every prime $p$.

This conjecture is the main theoretical motivation for the computational examples in the paper; the source points to the author's earlier paper for the conjecture and theoretical results. Its resolution is not specified in the supplied text.