Let $G$ a simple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.

Let $\lambda$ a minuscule or quasiminuscule weight.

For which types and for which weights do we have that: $\forall w\in W, \lambda-w\lambda$ is a multiple of a root?

For classical groups, we know that for types $A$, $B$, $C$ it's true for $\omega_{1}$ and true for $\omega_{1}$ and $\omega_{2}$ for $G_{2}$.

So the question concerns type $E$ and $F_{4}$.