Broadhurst–Kreimer dimension conjecture for multiple zeta values

Let $\mathcal Z_k$ be the space spanned by multiple zeta values of weight $k$, and let $\operatorname{gr}^{\operatorname{D}}_l\mathcal Z_k$ denote its depth-graded part of depth $l$. Define

$$\operatorname{\mathsf{E}}_2(x)=\frac{x^2}{1-x^2},\qquad \operatorname{\mathsf{O}}_3(x)=\frac{x^3}{1-x^2},\qquad \operatorname{\mathsf{S}}(x)=\frac{x^{12}}{(1-x^4)(1-x^6)}.$$

Broadhurst–Kreimer's conjecture. The generating series of the dimensions of the weight- and depth-graded parts is

$$\sum_{k,l\geq 0}\dim_{\mathbb Q}\left(\operatorname{gr}^{\operatorname{D}}_l\mathcal Z_k\right)x^ky^l=\frac{1+\operatorname{\mathsf{E}}_2(x)y}{1-\operatorname{\mathsf{O}}_3(x)y+\operatorname{\mathsf{S}}(x)y^2-\operatorname{\mathsf{S}}(x)y^4}.$$

This refines Zagier's weight-dimension conjecture by incorporating depth. The source attributes the refinement to Broadhurst and Kreimer and presents it as conjectural; no resolution is supplied.