Fuchs–Sanden conjecture for prime and twin-prime circles

Let $\mathcal P$ be a bounded primitive integral Apollonian packing. Define

$$\Pi_T(\mathcal P):=\#\{\text{prime }C\in\mathcal P:\operatorname{curv}(C)\le T\},\qquad \Pi_T^{(2)}(\mathcal P):=\#\{\text{twin primes }C_1,C_2\in\mathcal P:\operatorname{curv}(C_i)\le T\}.$$

Here $N_{\mathcal P}(T)$ denotes the relevant circle-counting function. Fuchs–Sanden conjecture. As $T$ tends to infinity,

$$\Pi_T(\mathcal P)\sim c_1\frac{N_{\mathcal P}(T)}{\log T},\qquad \Pi_T^{(2)}(\mathcal P)\sim c_2\frac{N_{\mathcal P}(T)}{(\log T)^2},$$

where $c_1>0$ and $c_2>0$ can be given explicitly. Upper bounds of the true order of magnitude are known, but the corresponding lower bounds remain open and very challenging.