Matching conjecture for the Stanley inequality graph

Let $\mathcal G=(V\sqcup W,E)$ be the bipartite graph defined above, with the parts and edges constructed from the relevant sets of linear extensions. Matching conjecture. There exists a maximal matching in $\mathcal G$ that covers every vertex in $W$. This conjecture proposes a matching-based route to a direct injection proving Stanley's inequality. The source notes that every vertex of $W$ has at least one incident edge, but does not establish the existence of a matching covering all of $W$.