Dimension conjecture for box-product spaces of qMZV words

For positive integers $z,d$, let $\mathscr{S}_{z,d}$ be the $\mathbb{Q}$-span of the box products $u_{\mathbf n}\boxast u_{\boldsymbol\ell}$ with $\operatorname{len}(\boldsymbol\ell)=d$ and total weight $z+d$, and write $\mathscr{s}_{z,d}=\dim_{\mathbb{Q}}\mathscr{S}_{z,d}$. Dimension conjecture. For all $z,d\in\mathbb{Z}_{>0}$,

$$\mathscr{s}_{z,d}=\binom{z+d-1}{\min\{z,d\}-1}.$$

The claim is based on numerical calculations in the paper and is presented as an open dimension formula for these box-product spaces.