Virtual-crystal realization conjecture for the four affine types

Let $4\mathfrak{g}4$ be of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, $A_{2n}^{(2)\dagger}$, or $C_n^{(1)}$, let $1\leq r\leq n$ and $s\geq1$, and let $V^{r,s}$ be the virtual crystal with extremal vector $u(V^{r,s})$. Let $4\mu_i4$ be defined by $4\mu_n=24$ for $4\mathfrak{g}=A_{2n}^{(2)\dagger}4$ and $4\mu_i=14$ otherwise. Virtual-crystal identification conjecture. $V^{r,s}$ is the simple crystal $B^{r,s}$ of the $U'_q(\mathfrak{g})$-module $W^{(r)}_s$, with extremal vector $u(V^{r,s})=u(\widehat{V}^{r,s})$ of weight $s\mu_r\overline{\Lambda}_r$; consequently it has the prescribed $U_q(\overline{\mathfrak{g}})$-decomposition and intrinsic energy $D_{V^{r,s}}=D_{B^{r,s}}$. This is the central conjectural identification behind the virtual-crystal construction; the paper proves related statements in special cases but does not establish it uniformly for all listed types and parameters.