The reducibility criterion for Weyl modules of type A

Let $G$ be a group of type $A$, let $\mathcal U^-$ denote the negative part of its enveloping algebra, and let $X^+(T)$ be the set of dominant weights. For a dominant weight $\omega$, write $e^+_\omega$ for a highest-weight vector in the corresponding Weyl module. A pair $(F,\omega)$ is reducible if it can be transformed to $(c,0)$ for some $c\in\mathbf K^*$ by the transformations defined in the paper.

The reducibility criterion. Let $F$ be a weight element of $\mathcal U^-$ and $\omega\in X^+(T)$. Then

$$Fe^+_\omega\ne0$$

if and only if the pair $(F,\omega)$ is reducible.

This criterion is proposed as a local characterization of nonzero vectors in Weyl modules for groups of type $A$. The supplied text does not indicate whether the conjecture has been resolved.