The arithmetic-progression symmetry conjecture for weighted finite multiple zeta sums

For $n_{1},\dots,n_{d}\in\mathbb{Q}$, define

$$W_{k}(n_{1},\dots,n_{d})=\sum_{\substack{k_{1},\dots,k_{d}\geq1\\ k_{1}+\cdots+k_{d}=k}}n_{1}^{k_{1}}\cdots n_{d}^{k_{d}}\zeta_{\mathcal{A}}(k_{1},\dots,k_{d}),$$

and let $W_{k}^{\star}(n_{1},\dots,n_{d})$ be the analogous sum with $\zeta_{\mathcal{A}}^{\star}$. For $k,r\in\mathbb{Z}_{\geq1}$ and $a,b\in\mathbb{Q}$, consider the arithmetic progression $(a,a+b,a+2b,\dots,a+rb)$. The arithmetic-progression symmetry conjecture.

$$W_{k}(a,a+b,a+2b,\dots,a+rb)=W_{k}(b,a+b,a+2b,\dots,a+rb),$$ $$W_{k}^{\star}(a,a+b,a+2b,\dots,a+rb)=W_{k}^{\star}(b,a+b,a+2b,\dots,a+rb).$$

This is presented as a conjectural weighted sum formula for finite multiple zeta values, distinct from the theorem earlier in the paper; its resolution is not indicated in the supplied text.