The Chevalley polytope Newton–Okounkov body conjecture

Let $\X=\G/\P$ be a homogeneous space with a projective embedding $\X\hookrightarrow\bP(V_\varpi)$, and let $\bs$ be a reduced expression for $\wP$. The Chevalley polytope Newton–Okounkov body conjecture. There exists at least one reduced expression $\bs$ such that $\cP_{\X,\varpi,\bs}$ is a Newton–Okounkov body for $\X$ with respect to the valuation $\nu_{\X,\varpi,\bs}$. Unlike in the minuscule case, some Chevalley polytopes may be strictly contained in the corresponding Newton–Okounkov body; the conjecture predicts that at least one reduced expression gives equality.