The prime-component counting conjecture for Apollonian packings

Let ${\mathcal{P}^{\operatorname{full}}}$ be a primitive integral Apollonian packing, and let $C_{{\mathcal{P}^{\operatorname{full}}}}(X)$ be the number of circles of curvature at most $X$. Let $N^{\operatorname{root}}_{{\mathcal{P}^{\operatorname{full}}}}(X)$ count prime component roots $(a,b,c,p)$ with $a,b,c\leq p\leq X$ and $p$ prime. By the paper's uniqueness result, this also counts prime components. Define

$$c=\prod_{\substack{p\ \text{prime}\p\equiv1\,(4)}}\frac{p^2}{p^2-1}\prod_{\substack{p\ \text{prime}\p\equiv3\,(4)}}\frac{p^2}{p^2+1}=0.9159\ldots,$$

and let $c”$ be a constant between $0$ and $2$. Prime-component counting conjecture.

$$N^{\operatorname{root}}_{{\mathcal{P}^{\operatorname{full}}}}(X)\sim c\frac{C_{{\mathcal{P}^{\operatorname{full}}}}(X)}{\log X}-c”\frac{C_{{\mathcal{P}^{\operatorname{full}}}}(X)}{(\log X)^2}\sim c\frac{C_{{\mathcal{P}^{\operatorname{full}}}}(X)}{\log X}.$$

The paper proves an upper bound of the order $C_{{\mathcal{P}^{\operatorname{full}}}}(X)/\log X$ and gives a heuristic justification for this asymptotic.