Connectivity conjecture for maximal weakly separated collections

Let $\operatorname{W}(k,n)$ be the set of maximal collections of pairwise weakly separated $k$-subsets of $[1\dots n]$, and let a $(2,4)$-move replace one of the two indicated members of a maximal collection by the other while preserving weak separability and maximality. Connectivity conjecture. For any collections $\mathcal{C},\mathcal{B}\in\operatorname{W}(k,n)$, there is a sequence of $(2,4)$-moves transforming $\mathcal{C}$ into $\mathcal{B}$. The source states this as a conjecture after proving that individual $(2,4)$-moves preserve weak separability and maximality; no general proof is supplied.