The conjecture on the uniform rigged configuration bijection

Let $B$ be a tensor product of Kirillov--Reshetikhin crystals of affine type. Let $\operatorname{RC}(B)$ be the corresponding set of rigged configurations, let $\eta$ interchange riggings and coriggings, let $D$ be the energy statistic on $B$, and let $R:B\to B'$ be the combinatorial $R$-matrix for a reordering $B'$ of the tensor factors. The recursively defined map $\widetilde{\Phi}:\operatorname{RC}(B)\to B$ is obtained from the maps $\widetilde{\delta}$, $\operatorname{lb}$, and $\operatorname{ls}$. Uniform bijection conjecture. The map $\widetilde{\Phi}$ is a classical crystal isomorphism satisfying

$$D\circ\widetilde{\Phi}\circ\eta=\operatorname{cc},$$

and the diagram expressing compatibility with $R$ commutes:

$$\begin{CD} \operatorname{RC}(B) @>{\widetilde{\Phi}}>> B\\ @V{\operatorname{id}}VV @VV{R}V\\ \operatorname{RC}(B') @>{\widetilde{\Phi}}>> B'. \end{CD}$$

This consolidates conjectures about the rigged configuration bijection, its statistics, and its compatibility with factor reordering; the paper proves the assertion in a substantial range of affine types, while the stated general form remains open.