Combinatorial R-matrix compatibility of the rigged-configuration bijection

Let $(\nu,J)\in\operatorname{RC}(B)$, where $B=B_1\otimes B_2$ is a two-fold tensor product of Kirillov--Reshetikhin crystals, and let $B'=B_2\otimes B_1$ be the tensor product with the factors reversed. Let $R$ denote the crystal isomorphism exchanging the two tensor factors. R-matrix compatibility conjecture. One has

$$R(\Phi_B(\nu,J))=\Phi_{B'}(\nu,J).$$

This asserts that the proposed bijection is compatible with the combinatorial R-matrix when two KR factors are interchanged; the supplied text does not give a resolution.