Local-global conjecture for reduced curvatures in K-Apollonian packings

Let $K$ be an imaginary quadratic field with discriminant $\Delta\neq -3$. Let $\mathcal P$ be a $K$-Apollonian packing. The curvature of a $K$-Bianchi circle has the form $n\sqrt{-\Delta}$ with $n\in\mathbb Z$; call $n$ its reduced curvature. For any $M\in\mathbb Z$, let $S_M$ be the set of residue classes modulo $M$ of the reduced curvatures in $\mathcal P$.

Local-global conjecture for $K$-Apollonian packings. There exists an $M$ dividing $24$ such that all sufficiently large integers whose residues modulo $M$ lie in $S_M$ occur as reduced curvatures in $\mathcal P$. Furthermore, the minimal sufficient $M$ is determined by

$$v_2(M)=\begin{cases}3&\Delta\equiv28~(\textup{mod}~32),\\2&\Delta\equiv8,12,20,24~(\textup{mod}~32),\\1&\Delta\equiv0,4,16~(\textup{mod}~32),\\0&\text{otherwise},\end{cases}\qquad v_3(M)=\begin{cases}1&\Delta\equiv5,8~(\textup{mod}~12),\\0&\text{otherwise}. \end{cases}$$

This extends the local-global principle for integral Apollonian packings to imaginary quadratic fields other than $\mathbb Q(\sqrt{-3})$, predicting that the stated congruence conditions account for all sufficiently large reduced curvatures. The source gives no evidence of a resolution.