Multiple Clausen and Glaisher values generated by shifted log-sine integrals

For $k,d\in\mathbb{Z}_{\geq 0}$, define

$$S^{o}_{k,d}=\left\{\pi^{m}\operatorname{SLs}(k_{1},\dots,k_{n})\mathrel{}\middle|\mathrel{}m+k_{1}+\dots+k_{n}=k,\ d\geq n\geq 0,\ n\text{ odd},\ m\geq 0,\ k_i\geq 2\right\},$$ $$S^{e}_{k,d}=\left\{\pi^{m}\operatorname{SLs}(k_{1},\dots,k_{n})\mathrel{}\middle|\mathrel{}m+k_{1}+\dots+k_{n}=k,\ d\geq n\geq 0,\ n\text{ even},\ m\geq 0,\ k_i\geq 2\right\}.$$

Let $\mathcal{C}_k$ and $\mathcal{G}_k$ be the spaces spanned by multiple Clausen and multiple Glaisher values of weight $k$, respectively. Generation conjecture for Clausen and Glaisher values. Every multiple Clausen value of weight $k$ and depth $d$ is a $\mathbb{Q}$-linear combination of elements of $S^{o}_{k,d}$, and every multiple Glaisher value of weight $k$ and depth $d$ is a $\mathbb{Q}$-linear combination of elements of $S^{e}_{k,d}$. The assertion is open.