The virtualization conjecture for Kirillov–Reshetikhin crystals

Let $B^{r,s}$ be a Kirillov--Reshetikhin crystal for an affine type, and let $\phi$ and $(\gamma_a)_{a\in I}$ be the folding and scaling data defining virtual crystals. Define

$$\widehat{B}^{r,s}=\begin{cases} B^{n,s}\otimes B^{n,s} & \text{if } \mathfrak{g}=A_{2n}^{(2)},A_{2n}^{(2)\dagger}\text{ and }r=n,\\ \displaystyle\bigotimes_{r'\in\phi^{-1}(r)}B^{r',\gamma_r s}&\text{otherwise}.\end{cases}$$

Virtualization conjecture. There exists a virtualization map from $B^{r,s}$ into $\widehat{B}^{r,s}$. Virtualization is the mechanism used to transfer crystal and rigged-configuration constructions between affine types; the source describes this as conjectural and does not establish it in complete generality.