The upper-ledge containment conjecture

Let $\Pi$ be the set of pairs of partitions, let $i\in\mathbb{Z}_2$, and let $\mathcal{X}_i:\Pi\rightarrow\mathcal{UL}$ be the associated map to upper ledge diagrams. If $\pi,\tilde{\pi}\in\Pi$ and $\tilde{\pi}$ is obtained by deleting the last row of one component of $\pi$, then upper-ledge containment conjecture.

$$\mathcal{X}_i(\tilde{\pi})\subseteq\mathcal{X}_i(\pi).$$

The paper states this as a consequence that would follow from the preceding partition-containment conjecture, linking multipartition operations to upper-ledge crystal combinatorics.