Integrality conjecture for the Painlevé I series coefficients

Let the series

be the series solution introduced in the paper, with coefficients $A_{\ell,m,n}$ indexed by nonnegative integers. **Integrality conjecture.** The series

has

$$A_{\ell,m,n}\in\mathbb{Z}\qquad\forall\ell,m,n\geq 0.$$

The conjecture is suggested by known integrality results for the coefficients $a_{m,n}$ in the expansion of the sigma function and analogous results for sigma functions of some higher-genus curves; its resolution is not established in the supplied text.