Bipartite matching complex homology tensor-product conjecture

Let ${\mathsf {BNM}}_k(r,s)$ be the bipartite matching complex on two parts of sizes $r$ and $s$, with the natural action of $S_r\times S_s$. Let $H(n,l)$ denote the irreducible $S_n$-module corresponding to the hook Young diagram with $l$ rows and $n-l+1$ columns. Bipartite homology conjecture. As an $(S_r\times S_s)$-module, the reduced homology group in dimension $2k-3$ is isomorphic to

$$\widetilde{H}_{2k-3}({\mathsf {BNM}}_k(r,s)) \cong H(r,k)\otimes H(s,k).$$

The source says that the complex has homology concentrated in a single dimension and that experiments together with rank computations suggest this module-theoretic identification. Its status is unresolved in the supplied source.