The weighted sum vanishing conjecture for odd repeated-terminal indices

Let $W_{k}(n_{1},\dots,n_{d})$ and $W_{k}^{\star}(n_{1},\dots,n_{d})$ denote the weighted sums of finite multiple zeta values and finite star multiple zeta values, respectively, defined by

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

For $k\in\mathbb{Z}_{\geq1}$ and a positive odd integer $r$, the index sequence is $(1,1,2,3,\dots,r-2,r-1,r,r)$. The weighted sum vanishing conjecture.

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

This is one of the paper's conjectural weighted sum formulas for finite multiple zeta values, and no resolution is supplied in the given text.