Cubical-distance conjecture for largest minors

Let $\mathcal{J}\subset {[n]\choose k}$ be an arrangement of largest minors, and let $W\in {[n]\choose k}$. Say that $W$ is $(\geq t,\mathcal{J})$-largest if, for every arrangement of minors ending at $\mathcal{J}$, it is absent from the final $t-1$ arrangements. For a maximal arrangement $\mathcal{J}$ of largest minors and $W\notin\mathcal{J}$, let $cube_d(W,\mathcal{J})$ denote their cubical distance. Cubical-distance conjecture. If

$$cube_d(W,\mathcal{J})=t,$$

then $W$ is $(\geq t+1,\mathcal{J})$-largest minor. This conjecture predicts that cubical distance from a maximal arrangement controls how far below the largest minors an omitted minor lies. Its resolution is not indicated in the supplied text.