The vanishing ratio conjecture for primes with fixed and diagonal indices

Let $p^{(k)}_n$ denote the $k$-th prime in the subsequence of primes having prime index, with $n$ specifying the subsequence, and let $p^{(k)}_k$ be the corresponding diagonal term. For every $n\in\mathbb N$, vanishing ratio conjecture.

$$\lim_{k\to +\infty}\frac{p^{(k)}_n}{p^{(k)}_k}=0.$$

This conjecture asserts that, despite the asymptotic equivalence of the associated counting functions, the individual terms with fixed index parameter $n$ are negligible compared with the diagonal terms as $k$ tends to infinity. Its resolution is not supplied in the source.