Brown's matrix conjecture for multiple zeta values

Let $C_{N,r}$ be the matrices defined from the matrices $E^{(i)}_{N,r}$, and let $\mathbb{O}(x)$ and $\mathbb{S}(x)$ denote the generating series appearing in the source. Brown's matrix conjecture. The ranks of the matrices $C_{N,r}$ satisfy

$$1+\sum_{N,r>0}\operatorname{rank} C_{N,r} x^{N}y^r=\frac{1}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2}.$$

This conjecture gives a generating-series prediction for the ranks of the depth-indexed matrices governing relations in the depth-graded setting. The supplied text gives no resolution status.