Motivic Broadhurst–Kreimer conjecture for the depth-graded multiple zeta algebra

Let $d4d5$ be the algebra of motivic multiple zeta values, with increasing depth filtration $d49f$, and write $\operatorname{gr}^{\mathcal{D}}_d\mathcal{H}_N$ for its depth-$d$, weight-$N$ graded piece. Define

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

Motivic Broadhurst–Kreimer conjecture. The depth-and-weight generating series satisfies

$$\sum_{N,d\geq 0}\dim_{\mathbb{Q}}\bigl(\operatorname{gr}^{\mathcal{D}}_d\mathcal{H}_N\bigr)s^dt^N =\frac{1+\mathbb{E}(t)s}{1-\mathbb{O}(t)s+\mathbb{S}(t)s^2-\mathbb{S}(t)s^4}.$$

This conjecture predicts the dimensions of all depth-graded pieces of the motivic multiple zeta algebra and was motivated by exhaustive numerical computations of Broadhurst and Kreimer. The source gives no resolution.