A note on $U_q(D_4^{(3)})$ - Demazure crystals

We show that there exists a suitable sequence $\{w^{(k)}\}_{k \ge 0}$ of Weyl group elements for the perfect crystal $B = B^{1,3l}$ such that the path realizations of the Demazure crystals $B_{w^{(k)}}(l\Lambda_2)$ for the quantum affine algebra $U_q(D_4^{(3)})$ have tensor product like structure with mixing index $\kappa =1$.