Jonsson's 3-torsion conjecture for matching complexes

Let $n\geq 1$ and write

$$n=2k+1+3r,\qquad d=k-1+r.$$

Thus the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$. Jonsson's 3-torsion conjecture. The group contains elements of order $3$ if and only if $k\geq 0$ and $r\geq 2$, and it is an elementary nonvanishing $3$-group if and only if $0\leq k\leq r-2$. This strengthens the known sufficient conditions for 3-torsion and elementary 3-groups in the parameter range, while the asserted equivalences remain conjectural.