The totally odd spanning conjecture for motivic multiple zeta values

Let ${\mathcal{D}}_r$ and ${\mathcal{B}}_r$ denote the depth and block filtrations, respectively, on the space of motivic multiple zeta values, and let ${\mathcal{L}}$ denote the relevant motivic Lie coalgebra. For depth $r$, consider the quotient

$$\frac{{\mathcal{D}}_r{\mathcal{L}}}{{\mathcal{D}}_r{\mathcal{L}}\cap{\mathcal{B}}_{r-1}{\mathcal{L}}}.$$

Totally odd spanning conjecture. Totally odd zeta values of depth $r$ span this vector space. The claim is a reformulation of the proposed identification between the totally odd part and the quotient of the depth filtration by the lower block filtration.