The general odd-prime torsion conjecture for matching complexes

Let $p=2q-1$ be an odd prime, and let $n=2k+1+3r$ and $d=k-1+r$. Then the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$. The general odd-prime torsion conjecture. The group contains elements of order $p$ whenever $r\geq q$ and

$$k\geq(q-2)r-\binom{q-1}{2}.$$

Equivalently, $\tilde{H}_d(\mathsf{M}_n;\mathbb{Z})$ contains elements of order $p$ whenever

$$\frac{(q-1)(n-q/2)}{2q-1}-1\leq d\leq\frac{n-q-3}{2}.$$

The source describes this as having even less evidence than the preceding prime-specific conjectures; its general validity remains open.