Path-independence conjecture for monotonicity embeddings

Let $R\trianglerighteq R'$ be a covering relation in the dominance order on sequences of rectangles. Let $\theta_R^{R'}:LRT(\lambda;R)\to LRT(\lambda;R')$ be the embedding obtained from rectangle switching. Let $\Phi_R$ be the rigged-configuration bijection. Monotonicity embedding conjecture. The map $\theta_R^{R'}$ is independent of the sequence of covering relations leading from $R$ to $R'$, and the diagram

$$\begin{CD} LRT(\lambda;R) @>{\theta_R^{R'}}>> LRT(\lambda;R') \\ @V{\Phi_R}VV @VV{\Phi_{R'}}V \\ RC(\lambda^t;R^t) @>{\operatorname{inclusion}}>> RC(\lambda^t;(R')^t) \end{CD}$$

commutes. This would make the tableau embeddings canonical and compatible with the evident inclusion of rigged-configuration sets.