The complementary-pole coefficient-product divisibility conjecture

Let $G_{k,D}^{(r)}$ be the CM higher-order-pole forms, let $k\in\{4,6,8,10,14\}$, $1\leq r\leq k-1$, and let $D<-4$ be a discriminant with class number $1$. The complementary-pole divisibility conjecture. For every $n\in\mathbb{Z}^+$,

$$n^{k-2}\mid a_n(G_{k,D}^{(r)})a_n(G_{k,D}^{(k-r)}).$$

This is a numerical divisibility phenomenon for complementary pole orders in the CM family and remains open.