Rigged-configuration bijection and energy conjecture for type D4^(3)

Let

$$B=\bigotimes_{i=1}^N B^{r_i,s_i}$$

be a tensor product of type $D_4^{(3)}$ KR crystals. Let $\operatorname{RC}(B)$ be the corresponding rigged configurations, let $\Phi$ be the recursively defined map to $B$, and let $\eta$ be the complement map on riggings. Rigged-configuration bijection conjecture. The map

$$\Phi:\operatorname{RC}(B)\longrightarrow B$$

is a bijection, and $\Phi\circ\eta$ sends cocharge to energy. This conjecture would give the full type $D_4^{(3)}$ rigged-configuration model and imply the corresponding $X=M$ identity on highest weight elements.