Homology concentration conjecture for matching complexes

Let $d>1$ be a positive integer and let $A$ be a finite set. The matching complex $C_A^d$ is the simplicial complex whose vertices are the $d$-element subsets of $A$ and whose faces consist of mutually disjoint such subsets. For $n=1,\ldots,3d-1$, consider the matching complex $C_{nd}^d$ on a set of cardinality $nd$, and write $\widetilde H_i$ for its reduced homology groups.

Matching-complex homology conjecture. The only non-zero homology group of $C_{nd}^d$ for $n=1,\ldots,3d-1$ is

$$\widetilde H_{n-2}(C_{nd}^d).$$

Via the correspondence between syzygies of Veronese embeddings and homology of matching complexes, this is presented as equivalent to the Ottaviani–Paoletti conjecture. Since the latter is proved for third Veronese embeddings in the source, this equivalent formulation is solved in the case $d=3$; the general statement is not resolved by the supplied text.