The kissing-prime counting conjecture for Apollonian circle packings

Let $P$ be an Apollonian circle packing. For $x>0$, let $N_P(x)$ be the number of circles in $P$ with curvature less than $x$. Let $\psi_P^{(2)}(x)$ be the weighted sum over unordered pairs $(C,C')$ of tangent circles in $P$ having prime curvature, with both curvatures less than $x$:

$$\psi_P^{(2)}(x)=\sum_{\substack{(C,C')\in S\a(C),a(C')<x}}\log(a(C))\log(a(C')),$$

where $S$ is the set of such pairs. Let $\chi_4$ be the character modulo $4$ with $\chi_4(p)=1$ for $p\equiv1\pmod4$ and $\chi_4(p)=-1$ for $p\equiv3\pmod4$.

Kissing-prime counting conjecture. As $x\to\infty$,

$$\psi_P^{(2)}(x)\sim c\,L^2(2,\chi_4)\,N_P(x),$$

where

$$c=2\prod_{p\equiv3\,(4)}\left(1-\frac{2}{p(p-1)^2}\right)=1.646\dots.$$

The conjecture concerns the asymptotic number of tangent pairs of prime-curvature circles, or kissing primes. The paper presents a heuristic based on treating the Möbius function as random and reports computational support, but no proof or resolution.