Conjecture on the dual equivalence graph inside the skeleton of a crystal

Let $\lambda$ be a partition, and let $Skeleton(\lambda)$ be the graph obtained from the crystal of tableaux $B(\lambda)$ by replacing each quasicrystal with its associated standard tableau and retaining the edges of minimal index. The dual equivalence graph containment conjecture. The unoriented and unlabelled graph structure of $Skeleton(\lambda)$ contains that of the dual equivalence graph for $\lambda$. In particular, if there are $r$ two-sided arrows between two standard tableaux in the dual equivalence graph, then there are also $r$ edges between the same tableaux in $Skeleton(\lambda)$; conversely, if $Skeleton(\lambda)$ has $r>1$ edges between two standard tableaux, then the dual equivalence graph has $r$ edges between them. This would identify the dual equivalence graph as a distinguished part of the skeleton and clarify the relation between crystal structure and dual equivalence.