Apéry constants of Fano manifolds

Let $X$ be a Fano manifold and let $\gamma\in H^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(X)$ be a homogeneous coprimitive cohomology class of codimension $n$ with respect to $-K_X$. Let $A_0$ and $A_\gamma$ be the solutions of the quantum $D$-module associated with $1$ and $\gamma$, respectively, and write their relevant coefficients as $a_0^{(k)}$ and $a_\gamma^{(k)}$. Let $R=\mathbb{Q}[c_1,c_2,c_3,\dots]$ with $\deg c_i=i$, and let $ev:R\to\mathbb{C}$ send $c_1$ to Euler's constant and $c_i$ to $\zeta(i)$ for $i\geqslant 2$. Apéry's conjecture for Fano manifolds. The Apéry number

$$\lim_{k\to\infty}\frac{a_\gamma^{(k)}}{a_0^{(k)}}$$

is equal to $ev(f_\gamma)$ for some homogeneous polynomial $f_\gamma\in R^{(n)}$ of degree $n$. This predicts that Apéry constants arising from quantum $D$-modules are polynomial expressions in Euler's constant and zeta values, and is the main conjecture verified in examples in the paper; its general validity is not established by the supplied text.