The CM magnetic divisibility conjecture for higher-order poles

Let $G_{k,D}^{(r)}$ denote the integral linear combination of $E_k/(j-j(\alpha_D))^i$ for $1\leq i\leq r$ having a pole of order $r$ at the CM point of discriminant $D$. Let $k\in\{4,6,8,10,14\}$, $1\leq r\leq k-1$, and set $r'=\min(r,k-r)$. Let $D<-4$ be a discriminant with class number $1$. The CM magnetic divisibility conjecture. The form $G_{k,D}^{(r)}$ is $(r'-1)$-magnetic: for every $n\in\mathbb{Z}^+$,

$$n^{r'-1}\mid a_n(G_{k,D}^{(r)}).$$

This predicts systematic divisibility of Fourier coefficients in the CM family; the source presents it as a numerical conjecture and does not establish it in general.