Local-global conjecture for curvatures in integral circle packings

Let $\mathcal{P}$ be an integral circle packing, let $\mathcal{K}\subset\mathbb{Z}$ be its set of curvatures, and let $\mathcal{K}_a$ be the set of integers passing all local obstructions for $\mathcal{K}$. For $N\geq 0$, write

$$\mathcal{K}(N)=\mathcal{K}\cap[0,N]$$

and let $c_{M,\mathcal{A},C}$ denote the proportion of admissible congruence classes, so that $\#\mathcal{K}_a(N)=c_{M,\mathcal{A},C}N+O(1)$. Local-global conjecture. All sufficiently large admissible integers are curvatures; equivalently,

$$\#\mathcal{K}(N)=c_{M,\mathcal{A},C}N+O(1).$$

The preceding local result shows that the admissible integers are a finite union of congruence classes and gives their asymptotic density. The conjecture asserts that there are no further global obstructions beyond these local ones, apart from finitely many small admissible integers.