Tensor-product realization conjecture for the virtual type D4^(1) crystal

Let $\widetilde{B}^{3,s}$ be the crystal represented by tableaux in a $3\times s$ rectangle with classical decomposition

$$\widetilde{B}^{3,s}\cong\bigoplus_{k=0}^s B\bigl(k\overline{\Lambda}_1+(s-k)(\overline{\Lambda}_3+\overline{\Lambda}_4)\bigr).$$

Let $\widehat{B}^{3,s}$ and $\widehat{B}^{4,s}$ be the relevant type $D_4^{(1)}$ KR crystals. Tensor-product realization conjecture. There is an isomorphism

$$\widetilde{B}^{3,s}\cong\widehat{B}^{3,s}\otimes\widehat{B}^{4,s}$$

as $U_q'(\mathfrak{g})$-crystals of type $D_4^{(1)}$. This identifies the tableau model with the virtual tensor product needed for the folding construction.