The mixing-index conjecture for affine Demazure modules

Let $B$ be a perfect crystal, let $\{w^{(k)}\}$ be a sequence of Weyl group elements, and let $\lambda=l\Lambda_0$ be the level-$l$ dominant integral weight for which the sequence is considered. The conditions (II), (III), (IV), and (II') are the conditions defined in the paper, with (II') requiring $B^{(j+1,j)}_d=B^{(j+1)}_d\otimes B$ for every $j\geq 1$. The mixing-index conjecture. If $\{w^{(k)}\}$ satisfies (II), (III), and (IV) for $\lambda=l\Lambda_0$, then, for any level-$l$ dominant integral weight $\lambda$, it also satisfies (II') and (III). This predicts that the mixing index is at most $2$ after changing the highest weight while keeping the sequence $\{w^{(k)}\}$ unchanged.