The local-global conjecture for Apollonian 3-circle packings

Fix a primitive Apollonian 3-circle packing $\mathcal{P}$, let $\mathcal{K}$ be its set of curvatures, and let $A_{\mathcal{P}}$ be the set of admissible integers, meaning integers that pass all local obstructions for $\mathcal{P}$.

Local-global conjecture. Every sufficiently large admissible integer from $\mathcal{P}$ is a curvature. Equivalently,

$$\#\{n\in\mathcal{K}\mid n\leq N\}=\#\{n\in A_{\mathcal{P}}\mid 0<n\leq N\}+O(1).$$

The conjecture asserts that the local congruence conditions determine all sufficiently large curvatures in an Apollonian 3-packing. The paper proves a density-one version of this statement; the full conjecture is left open in the supplied text.