The elementary-$3$ and cyclic-order conjecture for $D_n$ homology

Let $D_n$ be the graph considered in the paper, let $M(D_n)$ be its matching complex, and let $\nu_{2n}$ denote the relevant homology dimension. The preceding theorem shows that $\widetilde H_{\nu_{2n}}(M(D_n))\neq 0$ for $n\geq 3$.

The $D_n$ conjecture. There exists an integer $n_0\geq 8$ such that, if $n\geq n_0$, then $\widetilde H_{\nu_{2n}}(M(D_n))$ is an elementary $3$-group. Moreover, if $n\geq n_0$ and $n\equiv 2\pmod 3$, then

$$\widetilde H_{\nu_{2n}}(M(D_n))\cong\mathbb Z_3.$$

This conjecture is suggested by the computational data and the torsion results for chessboard complexes; it predicts both eventual elementary $3$-torsion and, in one congruence class, a single cyclic summand.