Batyrev–Borisov stringy E-function conjecture for Gorenstein polytopes

Let $\Delta$ be a Gorenstein polytope of CY-dimension $n$, so that its stringy E-function $E_{\rm st}(\Delta;u,v)$ and CY-dimension are defined as in the preceding discussion. Stringy E-function conjecture. The function

$$E:= E_{\rm st}(\Delta;u,v) = \sum_{p,q}(-1)^{p+q}h^{p,q}_{\Delta}u^pv^q$$

is a polynomial in $u,v$ of degree $2n$ with nonnegative integral coefficients $h^{p,q}_{\Delta}$. In particular, $E=0$ if $n<0$, and $E$ is constant if $n=0$. For any $n\geq 1$, it satisfies

$$E_{\rm st}(\Delta;u,0)=(-u)^nE_{\rm st}(\Delta;u^{-1},0)$$

and

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

Together with the symmetry and Poincare-duality properties stated earlier, these conditions imply that if $n=1$, then $E=k(1-u)(1-v)$ for some integer $k$, while if $n=2$, then

$$E=k(1+u^2)(1+v^2)-2l(u+v)(1+uv)+(20k-16l)uv$$

for some integers $k,l$. In particular, $E_{\rm st}(\Delta;1,1)=24(k-l)$ is divisible by $24$. This conjecture predicts polynomiality and nonnegative integral stringy Hodge-number coefficients; the stated symmetry, duality, and low-dimensional consequences are part of the same claim, while no resolution is supplied here.