Conjecture on Ehrhart quasipolynomials of half-integral polygons

Let $i\in\mathbb{Z}_{\geq 2}$ be the number of interior lattice points of a half-integral polygon, and require the polygon to have at least $2$ boundary lattice points. An Ehrhart quasipolynomial enumeration conjecture. There are exactly

$$\frac{9}{2}i^3+36i^2+\frac{175}{2}i+53$$

Ehrhart quasipolynomials of such half-integral polygons. The authors verified the formula through $i=16$; its validity for all $i\geq 2$ remains open.