The unbounded denominators conjecture for meromorphic modular forms

Let $k$ be a weight, let $\Gamma$ be a finite-index subgroup of $SL_2({\mathbb Z})$, and let $f$ be a meromorphic weight $k$ modular form for $\Gamma$ that is holomorphic on the upper half-plane $\frak H$. Suppose that the Fourier coefficients of $f$ are algebraic, and say that they have bounded denominators if some nonzero algebraic integer $c$ makes every $c\,a(n)$ algebraically integral.

Unbounded denominators conjecture. The Fourier coefficients of $f$ have bounded denominators if and only if $f$ is a cusp form for a congruence subgroup.

This conjecture extends the expected characterization of congruence forms by the bounded-denominator property to meromorphic modular forms; the source notes consequences for graded dimensions of rational, $C_2$-cofinite vertex operator algebras. The conjecture is presented as open in the source.