The cocharge-energy conjecture for type D rigged configurations

Let $(\nu,J)$ be a rigged configuration and let $b=\Phi(\nu,J)$ be its image under the bijection to the corresponding tensor product of type $D_n^{(1)}$ Kirillov--Reshetikhin crystals. Let $\operatorname{cc}(\nu,J)$ denote the cocharge statistic and let $\overline{D}(b)$ denote the corresponding energy statistic. Cocharge-energy conjecture. The bijection preserves these statistics:

$$\operatorname{cc}(\nu,J)=\overline{D}(b).$$

The statement is proved for the highest weight element of a single KR crystal $B^{r,s}$, but the supplied status evidence does not establish the conjecture in full generality.