The local-to-global principle for integral Apollonian circle packings

Let $P$ be an integral Apollonian circle packing, and let $P_{24}$ be the set of residue classes modulo $24$ represented by curvatures of circles in $P$.

Local-to-global principle for Apollonian circle packings. There exists $X_P\in\mathbb Z$ such that every integer $x>X_P$ whose residue modulo $24$ lies in $P_{24}$ is the curvature of a circle in $P$.

This conjecture predicts that the congruence obstructions modulo $24$ are the only obstructions to sufficiently large integers occurring as curvatures. It gives a precise form of the strong-density conjecture for integral Apollonian circle packings; the source reports experimental evidence but does not state a resolution.