The hook-tableau conjecture for recording tableaux of uncrowding

Let $\lambda$ be a partition, let $\mu\subseteq\lambda$, and let $\mathcal{H}_{\lambda/\mu}$ be the set of recording tableaux arising from the uncrowding map on edge labeled tableaux. Let $\widehat{s}_{\lambda}(\mathbf{x}\mid\mathbf{a})$ denote the tableau associated with the factorial Schur function, with the labels in the hook tableaux changed appropriately. Hook-tableau conjecture. When restricted to the tableau for $\widehat{s}_{\lambda}(\mathbf{x}\mid\mathbf{a})$, the set $\mathcal{H}_{\lambda/\mu}$ is the set of hook $\lambda/\mu$-tableaux, with an appropriate change of labels. This identifies the otherwise uncharacterized recording tableaux in the uncrowding crystal isomorphism with hook tableaux and would give a concrete description of the recording component in this setting.