Rectangle-reordering symmetry conjecture for LR tableaux

Let $R$ be a sequence of rectangles and let $u$ be a permutation of the rectangle indices. The bijection $u_R:LRT(\lambda;R)\to LRT(\lambda;uR)$ is defined by composing the adjacent rectangle-switching bijections. Let $\Phi_R$ be the rigged-configuration bijection. Rectangle-reordering symmetry conjecture. The diagram

$$\begin{CD} LRT(\lambda;R) @>{u_R}>> LRT(\lambda;uR) \\ @V{\Phi_R}VV @VV{\Phi_{uR}}V \\ RC(\lambda^t;R^t) @= RC(\lambda^t;u(R^t)) \end{CD}$$

commutes. Consequently, $u_R$ is independent of the chosen reduced word and the maps $u_R$ define an action of the symmetric group on the collection of LR tableaux. The source notes that a version for single-row rectangles was known.