Conjecture on 2-divisible sets of multiple harmonic sums

Let $d$ be a positive integer, let ${\vec{s}}\in{\mathbb N}^d$, and let $J({\vec{s}}|2)$ be the associated $2$-divisible set.

Conjecture on 2-divisible sets. For every positive integer $d$ and every ${\vec{s}}\in{\mathbb N}^d$,

$$J({\vec{s}}|2)=\{0\}.$$

The paper presents supporting computational data for this assertion and separately discusses primes $3$ and $5$. The supplied text gives no resolution.