Atkin–Swinnerton-Dyer integrality conjecture for vector-valued modular functions

Let $\vec v(q)$ be a vector-valued modular function with components $v_i(q)$, and write the Fourier expansion of $v_i(q)$ with leading exponent $m_0^{(i)}$ and Fourier coefficients $c_n^{(i)}$. Let $\Gamma(N)\subset SL(2,\mathbb{Z})$ be a principal congruence subgroup. Atkin–Swinnerton-Dyer integrality conjecture. If $m_0^{(i)}\in\mathbb{Q}$ and all Fourier coefficients $c_n^{(i)}$ are algebraic integers, then $v_i(q)$ is a modular function for a principal congruence subgroup $\Gamma(N)\subset SL(2,\mathbb{Z})$ for some $N$. This formulation is the componentwise version used in the paper and implies the congruence-subgroup modularity needed for the subsequent corollary. Its general validity is presented as conjectural.