Brown's uneven Broadhurst–Kreimer conjecture for totally odd motivic multiple zeta values

Let ${\mathcal{A}}_{odd,N}$ be the span of totally odd motivic multiple zeta values of weight $N$, and let ${\mathcal{D}}_r$ denote depth at most $r$. Define

$${\mathcal{O}}(s)=\frac{s^3}{1-s^2},\qquad {\mathcal{S}}(s)=\frac{s^{12}}{(1-s^4)(1-s^6)}.$$

Uneven Broadhurst–Kreimer conjecture. The bigraded dimensions of the totally odd part are given by

$$\sum_{N\geq 0,r\geq 0}\dim\frac{{\mathcal{D}}_r{\mathcal{A}}_{odd,N}}{{\mathcal{D}}_{r-1}{\mathcal{A}}\cap{\mathcal{D}}_r{\mathcal{A}}_{odd,N}}s^Nt^d=\frac{1}{1-{\mathcal{O}}(s)t+{\mathcal{S}}(s)t^2}.$$

The conjecture predicts the dimensions of depth-graded motivic multiple zeta values spanned by totally odd values; it is presented as a standard conjecture due to Brown and is connected to the homological Broadhurst–Kreimer conjecture.