Hess–Hirsch's homotopy-sphere conjecture for the reduced weak separation complex

For $2\leq k\leq n-2$, let $\widetilde{\Delta}^{Sep}_{k,n}$ be the reduced complex obtained from the weak separation complex of weakly separated $k$-subsets of $[n]$ by removing its universal cyclic-interval vertices. Hess–Hirsch's conjecture. For every $2\leq k\leq n-2$, the reduced complex $\widetilde{\Delta}^{Sep}_{k,n}$ is homotopy equivalent to the $(n-4)$-sphere. This conjecture is based on computer experiments and is presented as a special case of conjectures on the topology of weak separation complexes; the source gives no proof or resolution.