Conjecture on the homotopical depth of higher matching complexes

Let $M_{n-2}(K_n)$ be the matching complex of $(n-2)$-element matchings in the complete graph $K_n$, and let $M_{n-1}(K_{n,n})$ be the matching complex of $(n-1)$-element matchings in the complete bipartite graph $K_{n,n}$. The homotopical depth conjecture. The homotopical depth of $M_{n-2}(K_n)$ and $M_{n-1}(K_{n,n})$ is equal to the respective shifted connectivity degree. This extends known results on the homotopical depth of matching complexes and would determine these invariants for two higher matching complexes; the conjecture is presented as strongly believed in the source, with no resolution established here.