Sharpness conjecture for Newton–Okounkov bodies of Hilbert schemes on toric surfaces

Let $X$ be a smooth, projective, toric surface, let $X^{[n]}$ be its Hilbert scheme of $n$ points, and write every divisor on $X^{[n]}$ as $D_n+rE$, where $D$ is a torus invariant divisor on $X$ and $r\in\mathbb Z$. Let $P_D$ be the Newton polytope of $D$, and let $\overline{\Delta}(D_n+rE)$ denote the convex upper bound for the Newton–Okounkov body $\Delta(D_n+rE)$ given by the inequalities in Theorem 1.1.

Sharpness conjecture. If $X$ is $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, or a Hirzebruch surface, then

$$\Delta(D_n+rE)=\overline{\Delta}(D_n+rE)$$

for all divisors $D_n+rE\in\operatorname{Pic}(X^{[n]})$. The conjecture asserts that the upper bound is the exact Newton–Okounkov body in these cases. For most other smooth projective toric surfaces, the containment is strict; the claim is motivated by explicit computations for small $n$, while the general equality remains open.