Conjecture that the $(r-1)$-st almost totally odd part spans the full space

For integers $k>r\ge3$, let $\mathcal H^{al}_{k,r}$ be the $\mathbb Q$-vector space defined from products of depth-graded motivic multiple zeta values with one even index, and let $\mathcal H^{al,(r-1)}_{k,r}$ be the subspace spanned by the $(r-1)$-st almost totally odd motivic multiple zeta values of weight $k$ and depth $r$. Spanning conjecture. For $k>r\ge3$, one expects

$$\mathcal H^{al,(r-1)}_{k,r}=\mathcal H^{al}_{k,r}.$$

The equality is known in some low-depth cases discussed immediately before the conjecture, but the asserted equality for all $k>r\ge3$ is not established in the source.