Batyrev–Nill polynomiality conjecture for stringy E-functions

Let $P$ be a Gorenstein polytope of index $r$ and dimension $d$, and let $E_{st}(P;u,v)$ denote its stringy $E$-function. Batyrev–Nill polynomiality conjecture. The function $E_{st}(P;u,v)$ is a polynomial of degree $2(d+1-2r)$, or it is $0$; in particular, it is $0$ when $d+1<2r$. The paper proves the polynomiality assertion, which was the part of the conjecture addressed here; the stated degree and vanishing assertions are not established by the supplied context.