Coefficient-reflection conjecture for finite multiple harmonic sums

Let $m,n\in\mathbb{N}$, and let $\gb_a$ denote the $\mathcal{A}$-Bernoulli numbers. Write $a_1+\dots+a_l\vdash n$ for the summation condition used in the source, and let $C(a_1,\dots,a_l)$ be the coefficient function in the expansion of $R_n^{(m,1)}$. Coefficient-reflection conjecture. For all $m,n\in\mathbb{N}$, if

$$R_n^{(m,1)}=n!\sum_{1\le l\le n/3,\ 2\mid(n-l)}\ \sum_{a_1+\dots+a_l\vdash n}C(a_1,\dots,a_l)\gb_{a_1}\dots\gb_{a_l},$$

then

$$S_n^{(m,1)}=n!\sum_{1\le l\le n/3,\ 2\mid(n-l)}\ \sum_{a_1+\dots+a_l\vdash n}C(-a_1,\dots,-a_l)\gb_{a_1}\dots\gb_{a_l}.$$

This conjecture proposes that the formula for $S_n^{(m,1)}$ is obtained from that for $R_n^{(m,1)}$ by reflecting the arguments of the coefficient function. The supplied text gives no resolution status.