Saturation of Demazure characters in type E

Let $G$ be a simple algebraic group of type $E$, let $W$ be its Weyl group, let $w\in W$, and let $\lambda$ be dominant. Let $V_\lambda^w$ be the corresponding Demazure module, and let $\operatorname{char}(V_\lambda^w)$ denote its character. A character is saturated when its support has the saturation property defined in the paper.

Saturation conjecture. For every $w\in W$ and every dominant $\lambda$, $\operatorname{char}(V_\lambda^w)$ is saturated.

The conjecture applies to the exceptional types $E_6$, $E_7$, and $E_8$. The paper explains that its methods do not cover type $E$, while the analogous saturation property is confirmed computationally for the treated types, including $G_2$.