Evacuation–rigged-configuration compatibility conjecture

Let $\operatorname{ev}$ be evacuation on column-strict tableaux, restricting to a bijection $LRT(\lambda;R)\to LRT(\lambda;\operatorname{rev}(R))$. Let $\theta$ complement each rigging label with respect to its vacancy number, giving an involution on rigged configurations. Evacuation compatibility conjecture. The diagram

$$\begin{CD} LR(\lambda;R) @>\operatorname{ev}>> LR(\lambda;\operatorname{rev}(R)) \\ @V{\Phi_R}VV @VV{\Phi_{\operatorname{rev}(R)}}V \\ RC(\lambda^t;R^t) @>>\theta> RC(\lambda^t;\operatorname{rev}(R^t)) \end{CD}$$

commutes. This is a compatibility statement between tableau evacuation and complementation of riggings; the source introduces it as the next reduction of the rectangle-reordering conjecture.