Dyadic nondivisibility conjecture for multiple harmonic sums

Let $l$ be a positive integer and let ${\boldsymbol{s}}\in\mathbb N^l$. Define $J({\boldsymbol{s}}|2)$ to be the set of nonnegative integers $n$ for which the numerator of the multiple harmonic sum $H({\boldsymbol{s}};n)$ is divisible by $2$. Dyadic nondivisibility conjecture. For every ${\boldsymbol{s}}$, one has $J({\boldsymbol{s}}|2)=\{0\}$. This conjecture is motivated by the authors’ explicit dyadic calculations and would imply that no positive-index multiple harmonic sum has an even numerator; it remains open.