The Hauptmodul and monodromy-group conjecture for Mahler flows

Let $\Gamma^a$ be the congruence subgroup associated to the dessin for reflexive polygons with maximally tempered coefficients. Let $k$ be the parameter of the Newton polynomial, let $n$ be the exponent appearing in the proposed Hauptmodul, and let $\Gamma^b$ be a congruence subgroup. Let the corresponding Picard–Fuchs equation be the differential equation governing the relevant period.

Hauptmodul and monodromy-group conjecture. Then $k^n$ is a Hauptmodul for $\Gamma^b$,

$$\Gamma^a\leq\Gamma^b,$$

and $\Gamma^b$ is the monodromy group of the corresponding Picard–Fuchs equation.

The claim connects dessins, congruence subgroups, Hauptmoduln, and Picard–Fuchs monodromy. The paper motivates it through computed examples but does not establish it in general.