Shifted log-sine basis conjecture for multiple zeta values

Let $\mathcal{Z}_{k}$ be the $\mathbb{Q}$-vector space spanned by multiple zeta values of weight $k$, and let $S'_{k,k}$ be the subset of shifted log-sine values defined by

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

Shifted log-sine basis conjecture. The set of real numbers $S'_{k,k}$ is a basis of $\mathcal{Z}_{k}$. If true, this would imply Zagier's dimension conjecture for multiple zeta values; it is open.