Multiple zeta values generated by shifted log-sine integrals

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

$$S'_{k,d}=\left\{\pi^{2m}\operatorname{SLs}(k_{1},\dots,k_{n}) \mathrel{}\middle|\mathrel{} 2m+k_{1}+\dots+k_{n}=k,\ m\geq 0,\ d\geq n\geq 0,\ k_i\geq 3\text{ odd}\right\},$$

and let a multiple zeta value have weight $k$ and depth $d$. Generation conjecture. Every multiple zeta value with weight $k$ and depth $d$ can be written as a $\mathbb{Q}$-linear combination of elements of $S'_{k,d}$. This extends the verified low-weight calculations; the general generation statement is open.