Weak separation purity conjecture for compatibility complexes

Let $[m]=\{1,\ldots,m\}$, and let ${\bf m}^{(ws)}_{k \leq l}$ be the compatibility complex whose vertices are the subsets of $[m]$ having at least $k$ and at most $l$ elements, with weak separation as the compatibility rule. Weak separation purity conjecture. The complex ${\bf m}^{(ws)}_{k \leq l}$ is pure. This conjecture predicts that all maximal weakly separated collections subject to the size restriction have the same dimension, namely the dimension given by the preceding theorem.