Broadhurst–Kreimer–Brown conjecture on the depth-graded motivic multiple zeta values

Let $\mathcal{H}$ be the algebra of motivic multiple zeta values, equipped with its depth filtration $\mathfrak{D}_r\mathcal{H}$, and let

$$gr_r^{\mathfrak{D}}\mathcal{H}=\mathfrak{D}_r\mathcal{H}/\mathfrak{D}_{r-1}\mathcal{H}.$$

Write $gr_r^{\mathfrak{D}}\mathcal{H}_N$ for its weight-$N$ part. Define

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

Broadhurst–Kreimer–Brown conjecture. The generating series satisfies

$$1+\sum_{N,r>0}\operatorname{dim}_{\mathbb{Q}}(gr_r^{\mathfrak{D}}\mathcal{H}_N)x^Ny^r=\frac{1+\mathbb{E}(x)y}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2-\mathbb{S}(x)y^4}.$$

The cases $r=2$ and $r=3$ are known, while the higher-depth cases remain unknown. This conjecture predicts the depth structure of motivic multiple zeta values and is a motivic form of the Broadhurst–Kreimer depth conjecture.