Asymptotic distribution of prime terms in $s_k(n)$

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer $n$. For a fixed $n$, define $(k_N)_{N\geq 1}$ by requiring that $s_{k_N}(n)$ is the $N$th prime term of $(s_k(n))_{k\geq 0}$. Let

$$\lambda=\frac{n+\sqrt{n^2-4}}{2},$$

and let $\gamma$ be the Euler–Mascheroni constant. Asymptotic prime-distribution conjecture for $s_k(n)$. If $n\geq 3$ and $n\neq\mathcal{T}_p(j)$ for every prime $p$ and every integer $j\geq 3$, then, as $N\to\infty$,

$$\log\log s_{k_N}(n)\sim C N,$$

where

$$C=e^{-\gamma}\log\sqrt{\lambda}.$$

Numerical evidence for small values of $n$ supports the predicted linear behaviour and the constant, but the asymptotic remains unproved.