MTWZ finite multiple zeta value algebraicity conjecture

Let $\mathbb{N}$ denote the positive integers. For each $m,n\in\mathbb{N}$, let $R_n^{(m,1)}$ and $S_n^{(m,1)}$ be the finite multiple harmonic sum quantities defined in the source, and let $\mathcal{A}_1$ be the finite multiple zeta value algebra introduced above. The $\mathcal{A}$-Bernoulli numbers are the elements $\gb_k=\big(-B_{p-k}/k\pmod p\big)_{p>k}\in\mathcal{A}_1$. MTWZ's conjecture. For any $m,n\in\mathbb{N}$, both $R_n^{(m,1)}$ and $S_n^{(m,1)}$ are elements of the sub-algebra of $\mathcal{A}_1$ generated by the $\mathcal{A}$-Bernoulli numbers. This conjecture predicts that these finite sums can always be expressed algebraically in terms of finite analogues of Bernoulli numbers; the supplied text gives no resolution status.