Conjecture on fixed-descent subgraphs of the skeleton of a crystal

Let $\lambda$ be a partition, and let $Skeleton(\lambda)$ be the skeleton whose vertices are standard tableaux of shape $\lambda$. For each positive integer $s$, let $H_s$ be the induced subgraph whose vertices are the standard tableaux with descent compositions having $s$ parts. The fixed-descent skeleton conjecture. Each $H_s$ is either a disjoint union of singleton vertices, a disjoint union of chains, or a disjoint union of even cycles, with or without two extra attached vertices giving the sources and sinks. Multiple edges occur only between induced subgraphs associated to different descent-composition lengths. This predicts a restricted and structured form for the parts of the skeleton obtained by fixing the number of descents.