The prime-indexed Abelian limit conjecture

Let $f(n)$ be an arithmetic function, let $p_n$ denote the $n$th prime, and let $\operatorname{idx}(p_n)=n$. Suppose that the Cesàro mean

$$\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}f(k)=L_f$$

exists. Prime-indexed Abelian limit conjecture. Then

$$\lim_{s\to 1^+}\frac{\sum_{p\in\mathbb P}f\left(\operatorname{idx}(p)\right)p^{-s}}{\sum_{p\in\mathbb P}p^{-s}}=L_f.$$

This extends the analogy between Abelian limits for power series and prime-indexed Dirichlet series; the indicator-function case recovers the known equality between Dirichlet and natural densities for subsets of the primes. The general assertion is presented as speculative and no resolution is supplied.