Conjecture on mean values of arithmetic subsequences of Hardy–Littlewood constants

Let $C_{2r}$ denote the Hardy–Littlewood constant associated with the prime pair $(p,p+2r)$. An arithmetic subsequence of the index sequence $\{2r\}$ means a subsequence obtained by restricting $2r$ to an arithmetic progression. Arithmetic-subsequence mean-value conjecture. Every subsequence of $\{C_{2r}\}$ corresponding to an arithmetic subsequence of $\{2r\}$ has a mean value. The text gives the mean value for subsequences indexed by $2hr$ and presents the general assertion as plausible; no resolution is stated.