Algebraic CM-supported residues of magnetic modular forms

Let $f$ be a magnetic modular form of weight $k$. For each integer $m$ with $0\leq m\leq k-2$, consider the residues of $\tau^m f(\tau)$ at the relevant points. A point in the upper half-plane is a complex multiplication point if it corresponds to a CM elliptic curve.

Residue conjecture. The residues of $\tau^m f(\tau)$ satisfy

$$\operatorname{Res}\bigl(\tau^m f(\tau)\bigr)\in \frac{1}{(2\pi i)^{k/2}}\overline{\mathbb{Q}},$$

and they are nonzero only at complex multiplication points.

The conjecture is based on computations of residues for many examples and candidates. It predicts both an algebraicity statement and a strong restriction on the locations where residues can occur.