Admissibility and Weyl-orbit conjecture for soliton contents

Let $B=B^{r_1,l_1}\otimes\cdots\otimes B^{r_L,l_L}$ be a tensor product of crystals of type $A_n$, and let $m=(m^{(a)}_j)$ be a soliton content. Define

$$P(m)=\{p\in B\mid p\text{ is evolvable},\ E^{(a)}_j(p)=\sum_{k\geq1}\min(j,k)m^{(a)}_k\},$$

let $p^{(a)}_j$ be the corresponding vacancy numbers, let $H=\{(a,j)\mid1\leq a\leq n,\ j\in\mathbb Z_{\geq1},\ m^{(a)}_j>0\}$, and let $\lambda(m)=\sum_{a=1}^n p^{(a)}_\infty\Lambda_a$. Admissibility and Weyl-orbit conjecture. One has

$$P(m)\neq\emptyset\quad\text{if and only if}\quad p^{(a)}_j\geq0\quad\text{for all }(a,j)\in H,$$

and

$$\{\operatorname{wt}p\mid p\in P(m)\}=W\lambda(m),$$

where $W$ is the Weyl group of $A_n$. This predicts both the vacancy-number criterion for nonemptiness and the exact Weyl-group orbit of the weights of the corresponding states. The supplied text gives no resolution status.