Tropical reflection agrees with evacuation of tableaux

Let $T\in\operatorname{SSYT}(k,[n])$, let $\theta$ be the tropical reflection map, let $\tau$ be the tropical twist map, let $\operatorname{ch}(T)$ be the associated dual canonical basis element, and let $\operatorname{eva}(T)$ be tableau evacuation.

Evacuation conjecture. For every $T\in\operatorname{SSYT}(k,[n])$,

$$\theta(\operatorname{ch}(T))=\operatorname{ch}(\operatorname{eva}(T)),\qquad \theta(T)=\operatorname{eva}(\tau^{-1}(T)).$$

The conjecture connects tropical reflection with Schützenberger evacuation, both on dual canonical basis elements and on tableaux. The supplied text gives no resolution status.