Normality conjecture for toric orbit closures in generalized flag varieties

Let $G$ be a semisimple algebraic group, let $P_{\lambda}$ be the parabolic subgroup associated with a dominant weight $\lambda$, and let $G/P_{\lambda}$ be embedded by the corresponding representation. For a general element of $G/P_{\lambda}$, its toric orbit closure is normal; equivalently, the associated semigroup $\mathbb{N}A_0$ is defined by inequalities determined by the support of $\lambda$ along the Dynkin diagram. Normality conjecture. The toric orbit closure of a general element in $G/P_{\lambda}$ is normal. The claim extends the established fundamental-weight cases and proposes that normality follows from the Dynkin-diagram description of the defining inequalities.