Antisymmetry conjecture for the matching preorder

Let $\mathcal{M}_n$ be the set of labelled perfect matchings, equipped with the preorder $\preceq$. Antisymmetry conjecture. The preorder $(\mathcal{M}_n,\preceq)$ is a poset; equivalently,

$$M\preceq N\text{ and }N\preceq M\quad\Longrightarrow\quad M=N.$$

The paper presents this as a consequence that would follow from the switch-generation conjecture, and the supplied parser gives no evidence that it has been resolved.