Reflexive polygon conjecture for infinite staircases in rational convex toric domains

A rational convex toric domain is a symplectic domain whose moment polygon is a rational convex polygon, and its ellipsoid embedding function records the minimum scaling factor needed to symplectically embed a four-dimensional ellipsoid into the domain. A convex lattice polygon is reflexive if it has exactly one interior lattice point.

Reflexive polygon conjecture. If the ellipsoid embedding function of a rational convex toric domain has an infinite staircase, then its moment polygon is a scaling of a reflexive polygon.

The conjecture asserts that the twelve reflexive polygons identified in the paper give, up to scaling and integral affine equivalence, all possible rational convex toric domains with infinite staircases; four other reflexive polygons are described as not producing infinite staircases. The paper reduces the exhaustiveness question to a number-theoretic problem related to work of Hardy and Littlewood.