Mason's bounded-denominator conjecture for vector-valued modular forms

Let $\Gamma=SL_2(\mathbb{Z})$, let $\rho$ be a representation of $\Gamma$, and let $F$ be a vector-valued modular form for $\rho$. Say that the components of $F$ have Fourier expansions with bounded denominators if the denominators of their Fourier coefficients are bounded, after the usual normalization. Mason's conjecture. If the components of $F$ have Fourier expansions with bounded denominators, then

$$\ker\rho\text{ is a congruence subgroup of }\Gamma.$$

This is a vector-valued generalization of the unbounded-denominators conjecture for classical modular forms. The paper presents it as a conjecture motivated by Mason's two-dimensional results; the cited work establishes substantial bounded-denominator results, but the general statement remains open.