$p$-Integrality of canonical coordinates

Let $L$ be a differential operator with coefficients in $\mathbb{Q}(z)$ of order $n\ge 2$ with maximal unipotent monodromy at zero. In this paper we are interested in determining when the canonical coordinate of $L$ belongs to ${\mathbb{Z}}_p[[z]]$. For this purpose, motivated by a recent conjecture due to P. Candelas, X. de la Ossa and D. van Straten~\cite{CD}, we study the situation when $L$ has a strong Frobenius structure $\mathrm{\Phi }=({\phi }_{i,j}{)}_{1\le i,j\le n}\in M_n({\mathbb{Z}}_p[[z]])$ such that ${\phi }_{1,1}(0)=1$. We then give a necessary and sufficient condition for the canonical coordinate of $L$ to belong to ${\mathbb{Z}}_p[[z]]$ when $L$ has such a strong Frobenius structure.