Duality compatibility conjecture for LR and rigged-configuration bijections

Let $R$ be a sequence of rectangles, let $\widetilde{R}$ and $\widetilde{\lambda}$ be the complementary dual data defined in the source, and let $\operatorname{dual}$ be the corresponding duality bijection on LR tableaux. Duality compatibility conjecture. The diagram

$$\begin{CD} LRT(\lambda;R) @>\operatorname{dual}>> LRT(\widetilde{\lambda};\widetilde{R}) \\ @V{\Phi_R}VV @VV{\Phi_{\widetilde{R}}}V \\ RC(\lambda^t;R^t) @>>> RC(\lambda^t;(\widetilde{R})^t) \end{CD}$$

commutes, where the bottom map is the previously defined rigged-configuration duality map. The source derives the corresponding equality of charge-generating polynomials before stating this compatibility conjecture.