Strong tableaux and Hikita tableaux equivalence conjecture

Let $\mathbf{m}\in\mathbb{E}_n$, let $P=P_{\mathbf{m}}$ be the associated natural unit interval order, and let $\lambda\vdash n$. Let $\operatorname{strongST}_P(\lambda)$ be the strong standard $P$-tableaux of shape $\lambda$, and let $\operatorname{HikSYT}(\mathbf{m},\lambda)$ be the Hikita tableaux of shape $\lambda$. Strong-tableau–Hikita equivalence conjecture. The set $\operatorname{strongST}_P(\lambda)$ is non-empty if and only if $\operatorname{HikSYT}(\mathbf{m},\lambda)$ is non-empty. The source states that this is equivalent to the strong-tableau nonvanishing conjecture in the natural-unit-interval case.