The corrected local-to-global conjecture for Apollonian curvatures

Let ${\mathcal{P}^{\operatorname{full}}}$ be a primitive integral packing, and let $\Sigma$ be the union of the residue classes modulo $24$ that have representatives in ${\mathcal{P}^{\operatorname{full}}}$. Let $S$ be the union of the curvature families ruled out by reciprocity obstructions catalogued in HKRS, Theorem 2.5. Corrected local-to-global conjecture. Every sufficiently large integer $m$ with $m\in\Sigma\setminus S$ occurs as a curvature in ${\mathcal{P}^{\operatorname{full}}}$.