Conjectural structure of the reserved sets RJ for multiple harmonic sums

Let ${\vec{s}}\in{\mathbb N}^l$. Write $RJ({\vec{s}})$, $RJ_1({\vec{s}})$, and $RJ_2({\vec{s}})$ for the reserved sets associated with the multiple harmonic sum indexed by ${\vec{s}}$.

Reserved-set structure conjecture. If one of the following holds: (i) ${\vec{s}}=1$, (ii) ${\vec{s}}=(1,2,1)$, (iii) ${\vec{s}}=(2r-1,1)$ for some $r\geq1$, or (iv) ${\vec{s}}=1^{2l}$ for some $l\geq1$, then

$$RJ({\vec{s}})=RJ_2({\vec{s}}).$$

For all other ${\vec{s}}$,

$$RJ({\vec{s}})=RJ_1({\vec{s}}).$$

The source attributes this conjecture to Conjecture 7.4 of the author's earlier paper. The supplied text gives no resolution.