The weighted sum vanishing conjecture for the index sequence $(1,1,2,3,\dots,r)$

Define the weighted sums of finite multiple zeta values by

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

and define $W_{k}^{\star}(n_{1},\dots,n_{d})$ similarly with $\zeta_{\mathcal{A}}^{\star}$ in place of $\zeta_{\mathcal{A}}$. For $k,r\in\mathbb{Z}_{\geq1}$, the index sequence is $(1,1,2,3,\dots,r-1,r)$. The weighted sum vanishing conjecture.

$$W_{k}(1,1,2,3,\dots,r-1,r)=0,$$ $$W_{k}^{\star}(1,1,2,3,\dots,r-1,r)=0.$$

These are conjectural weighted sum formulas for finite multiple zeta values, distinct from the theorem proved earlier in the paper; their status is not resolved in the supplied text.