The lower-depth totally odd spanning conjecture

Let ${\mathcal{D}}_n$ and ${\mathcal{B}}_{n-1}$ denote the depth and block filtrations, and work modulo products. Lower-depth totally odd spanning conjecture. The vector space

$$\frac{{\mathcal{D}}_n}{{\mathcal{D}}_n\cap{\mathcal{B}}_{n-1}}$$

is spanned by totally odd zeta values of depth $n$. This is presented as an equivalent-type reformulation of the depth-to-even-block isomorphism conjecture and concerns whether totally odd values generate the indicated depth quotient.