On the integrality of the Taylor coefficients of mirror maps, II

We continue our study begun in "On the integrality of the Taylor coefficients of mirror maps" (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. More precisely, we address the question of finding the largest integer $v$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/v}$ are still integers. In particular, we determine the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(G(z)/F(z))$, $F(z)=\sum_{m=0}^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1}^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.