Proof of Chapoton's conjecture on Newton polygons of $q$-Ehrhart polynomials

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial of an order polytope. In this paper, we prove Chapoton’s conjecture.