Switch-generation conjecture for the matching preorder

Let $\mathcal{M}_n$ be the set of the labelled perfect matchings under consideration, and let $\preceq$ be the preorder on this set. A switch is the local transformation defined earlier in the paper. Switch-generation conjecture. If $N,M\in\mathcal{M}_n$ satisfy $N\preceq M$, then there exists a sequence

$$N=M_1\preceq M_2\preceq\dots\preceq M_{k-1}\preceq M_k=M$$

such that $M_j$ is a switch of $M_{j+1}$ for every $j$. This is presented as an open conjecture about whether the preorder is generated by switches; the supplied parser gives no evidence of resolution.