Statistic-preserving rigged-configuration bijection conjecture

Let $R$ be a dominant sequence of rectangles. Let $LRT(\lambda;R)$ be the corresponding Littlewood–Richardson tableaux, let $RC(\lambda^t;R^t)$ be the rigged configurations, and let

$$\Phi_R:LRT(\lambda;R)\longrightarrow RC(\lambda^t;R^t)$$

be the stated bijection. Statistic-preserving bijection conjecture. For $T\in LRT(\lambda;R)$ and $(\nu,L)=\Phi_R(T)$,

$$\operatorname{charge}_R(T)=\operatorname{cocharge}(\nu,L).$$

This conjecture would explain the agreement between tableau charge and rigged-configuration cocharge, and would imply the corresponding generating-function identity.