The 3-rank conjecture for matching complexes

For $k\geq 0$, consider the reduced homology group $\tilde{H}_{k+1}(\mathsf{M}_{2k+7};\mathbb{Z})$, where $\mathsf{M}_n$ denotes the matching complex on $n$ vertices. The 3-rank conjecture. The group is the direct sum of a free group and an elementary $3$-group of rank

$$\binom{2k+6}{k}.$$

The conjecture refines the known examples of 3-torsion in matching-complex homology by predicting its exact rank; no resolution is supplied in the source.