Batyrev–Nill conjecture on Calabi–Yau dimensions of stringy E-functions

Let $P$ be a Gorenstein polytope of Calabi–Yau dimension $n$, so that $E_{st}(P;u,v)$ is its stringy $E$-function. Batyrev–Nill conjecture. The following assertions hold:

1. $E_{st}(P;u,v)$ is a polynomial.
2. It has degree $2n$ or is $0$; in particular, it is $0$ if $n<0$.
3. For $n\geq1$,

$$E_{st}(P;u,0)=(-u)^nE_{st}(P;u^{-1},0).$$

$$\left.\frac{d}{du}E_{st}(P;u,1)\right|_{u=1}=\frac{n}{2}E_{st}(P;1,1).$$

$$\left.\frac{d^2}{du^2}E_{st}(P;u,1)\right|_{u=1}=\frac{n(3n-5)}{12}E_{st}(P;1,1).$$

The first assertion is proved in the paper, while the supplied context does not say which of the remaining assertions are resolved.