Virtualization conjecture for type $D_4^{(3)}$ KR crystals

Let $\mathfrak{g}$ be of type $D_4^{(3)}$, and let $\widehat{\mathfrak{g}}$ be the folded type $D_4^{(1)}$. Write $\widehat{B}^{r,s}$ for KR crystals of type $D_4^{(1)}$. KR virtualization conjecture. There are virtualizations

$$B^{1,s}\longrightarrow \widehat{B}^{2,s},$$ $$B^{2,s}\longrightarrow \widehat{B}^{1,s}\otimes\widehat{B}^{3,s}\otimes\widehat{B}^{4,s}.$$

These embeddings realize the relevant type $D_4^{(3)}$ KR crystals inside crystals of type $D_4^{(1)}$; the conjecture is presented as a special case of a broader virtualization conjecture.