Box-kernel conjecture for qMZV box products

For positive integers $z,d$, let $\mathcal{K}_{z,d}$ be the explicitly defined subspace of the domain of the box map $\operatorname{Box}_{z,d}$ generated by the stuffle–box-product differences appearing in the source. Box-kernel conjecture. If $z\leq d$, then

$$\operatorname{span}_{\mathbb{Q}}\mathcal{K}_{z,d}=\operatorname{ker}\operatorname{Box}_{z,d}.$$

The inclusion from left to right is proved, while the converse is supported by numerical calculations and remains open in general.