The prime and thickened prime component conjecture for Apollonian packings

Let ${\mathcal{P}^{\operatorname{full}}}$ be an ambient primitive integral Apollonian packing. A prime component ${\mathcal{P}^{\operatorname{pr}}}$ is a component consisting of the prime-curvature circles specified in the paper, and a thickened prime component ${\mathcal{P}^{\operatorname{th}}}$ is obtained from a prime component by adding every circle tangent to it. Prime and thickened prime component conjecture.

1. The set of curvatures appearing in ${\mathcal{P}^{\operatorname{pr}}}$ is the same as the set of primes appearing in ${\mathcal{P}^{\operatorname{full}}}$, up to finitely many exceptions.
2. The set of curvatures appearing in ${\mathcal{P}^{\operatorname{th}}}$ is the same as the set of curvatures appearing in ${\mathcal{P}^{\operatorname{full}}}$, up to finitely many exceptions.

This is proposed as the analogue of the corrected curvature conjecture for components; the paper states that it remains open.