The permutation-equivariance conjecture for the rigged-configuration bijection

Let $B = \bigotimes_{k=1}^N B^{r_k,s_k}$ and let $B' = \bigotimes_{k=1}^N B^{r'_k,s'_k}$ be any reordering of the same tensor factors. Let $\operatorname{RC}(B)$ and $\operatorname{RC}(B')$ be the corresponding rigged-configuration models, let $\Phi$ be the crystal bijection, and let $R\colon B\to B'$ be the crystal isomorphism induced by the reordering.

Permutation-equivariance conjecture. The diagram relating $\operatorname{RC}(B)$ to $\operatorname{RC}(B')$ by the identity and $B$ to $B'$ by $R$, with vertical maps given by $\Phi$, commutes for any reordering $B'$.

This asserts that the rigged-configuration bijection is compatible with reordering tensor factors. It is stated conjecturally in the supplied source, with no resolution indicated.