Basis conjecture for the box-kernel relations

For positive integers $z,d$ with $z\leq d$, let $\mathcal{K}_{z,d}$ and $\operatorname{Box}_{z,d}$ be as defined in the source. Basis conjecture. If $z\leq d$, then $\mathcal{K}_{z,d}$ is a basis of $\operatorname{ker}\operatorname{Box}_{z,d}$. This refines the box-kernel conjecture by asserting linear independence as well as spanning; the paper gives no proof in general.