Cunningham chain counting heuristic

A Cunningham chain of length $k$ is a prime sequence of either first-kind form $\{p,2p+1,4p+3,\ldots,2^{k-1}p+2^{k-1}-1\}$ or second-kind form $\{p,2p-1,4p-3,\ldots,2^{k-1}p-2^{k-1}+1\}$. Let

$$B_k=2^{k-1}\prod_{p>2}\frac{p^k-p^{k-1}\min(k,\operatorname{ord}_p(2))}{(p-1)^k}.$$

Cunningham chain conjecture. The number of Cunningham chains of length $k$ beginning with a prime $p\leq N$ is approximately

$$B_k\int_2^N\frac{dx}{\log x\log(2x)\cdots\log(2^{k-1}x)}\sim\frac{B_kN}{(\log N)^k}.$$

This is a heuristic prediction for both kinds of Cunningham chains and remains open in general.