Unbounded denominators conjecture for noncongruence modular forms

Let $\Gamma\leq\operatorname{SL}_2(\mathbb{Z})$ be a finite-index noncongruence subgroup, and let $f$ be a noncongruence modular form for $\Gamma$ whose Fourier coefficients lie in $\overline{\mathbb{Q}}$.

Unbounded denominators conjecture. The Fourier coefficients of $f$ have unbounded denominators. Equivalently, for every positive integer $n$, the Fourier coefficients of $nf$ are not all algebraic integers.

This folklore conjecture distinguishes noncongruence modular forms from congruence modular forms, which have bases with algebraic-integral Fourier coefficients. The conjecture was proved by Calegari, Dimitrov, and Tang using Nevanlinna theory, although their proof does not determine which primes can occur in the unbounded denominators.