Integrality Conjecture for vector-valued modular forms

Let $f_i(\tau)$ be a component of a vector-valued modular form with Fourier expansion

$$f_i(\tau)=q^{n^{(i)}_0}\sum_{n\geq 0}a_n^{(i)}q^n,$$

where $n^{(i)}_0=p_i/N_i$ and $(p_i,N_i)=1$. Suppose that all coefficients $a_n^{(i)}$ are algebraic integers. Integrality Conjecture. Then $f_i(\tau)$ is a modular form for $\Gamma(N_i)$. Equivalently, in the representation-theoretic formulation, an integral vector-valued modular form transforming in a representation $\rho$ should have $\operatorname{Ker}(\rho)$ containing a principal congruence subgroup $\Gamma(N)$. The result is stated to be known for two- and three-dimensional vector-valued modular forms and proven for vector-valued modular functions arising from rational conformal field theory characters; the general claim remains open.