The local-global conjecture for curvatures in an integral Apollonian gasket

Let $\sG$ be the fixed primitive integral Apollonian gasket, let $\sB=\sB(\sG)$ be the set of its circle curvatures,

$$\sB:=\{n\in\Z: \exists C\in\sG\text{ with }b(C)=n\},$$

and call an integer represented if it lies in $\sB$. Let $\sA=\sA(\sG)$ be the set of admissible integers, meaning those that are everywhere locally represented:

$$n\in\sB\pmod q,\quad \forall q\geq 1.$$

For the gasket considered here, the admissible residue classes modulo $24$ are $2,3,6,11,14,15,18,$ and $23$. Local-global conjecture. Every sufficiently large admissible number is the curvature of some circle in $\sG$. This conjecture asserts that, apart from finitely many exceptions, the local congruence conditions completely characterize the curvatures occurring in the gasket. The local obstructions are explicit, but the asserted global representation statement is not established in the source.