The reflexive polygon conjecture for rational convex toric domains

Let $\Omega\subset\mathbb{R}^2$ be a rational convex toric domain, meaning that $\Omega$ is a convex connected open subset of the first quadrant containing the origin and having rational vertices. Let $c_\Omega(a)$ denote its ellipsoid embedding function. A convex polygon with integral vertices is reflexive if its dual polygon is also integral.

Reflexive polygon conjecture. The embedding function $c_\Omega(a)$ has infinitely many singular points only if some scaling of $\Omega$ is reflexive.

This conjecture generalizes the phenomenon established in the paper's special case: infinitely many singular points of the embedding function should be constrained by an arithmetic reflexivity condition on the defining polygon.