Lower-bound conjecture for the length of Cunningham chains

Let $l(p)$ be the length of the Cunningham chain generated by the prime $p$, and let $\mathbb P$ denote the set of primes. Cunningham-chain length conjecture. On the primes,

$$l(p)=\Omega\left(\frac{\log p}{\log\log p}\right).$$

Equivalently, the chain lengths attain this order of growth along the primes. The paper derives this conjecture heuristically from the expected distribution of Cunningham chains and the resulting behavior of the maximal length $k(N)$; it remains unproved.