Universal limiting pair correlation conjecture for Apollonian gaskets

Let $\mathcal{P}$ be an Apollonian gasket and let $\mathcal R\subset\mathbb C$ satisfy $\mu(\mathcal R)>0$. For $T>0$, let $F_{T,\mathcal R}(s)$ denote the pair correlation function defined in the source, and write $F_{T,\mathcal R}$ for this function of $s$. Universal limiting pair correlation conjecture. There exists a non-negative, monotone, continuously differentiable function $F$ on $[0,\infty)$, supported away from $0$, such that

$$\lim_{T\rightarrow\infty}F_{T,\mathcal R}(s)=F(s)$$

for every $s\in[0,\infty)$. Moreover, $F$ is independent of the chosen Apollonian gasket. The conjecture is motivated by numerical plots suggesting convergence, continuity of the derivative, and universality across different Apollonian gaskets; no proof or resolution is supplied here.