The 7-torsion conjecture for matching complexes

Let $n=2k+1+3r$ and $d=k-1+r$, so that the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$. The 7-torsion conjecture. The group contains elements of order $7$ whenever $r\geq 4$ and $k\geq 2r-3$. Equivalently, $\tilde{H}_d(\mathsf{M}_n;\mathbb{Z})$ contains elements of order $7$ whenever

$$\frac{3n-13}{7}\leq d\leq\frac{n-7}{2}.$$

The source notes that the available evidence is limited, so this remains an open conjectural extension of the known cases.