The rogue divisor conjecture for Cunningham chains

Let $\mathbb{P}$ denote the primes, let $p\in\mathbb{P}$ with $p>3$, and let $C_i[p]$ be the Cunningham chain associated with $p$ and index $i$. Define

$$k=2\max\{C_i[p]\}+(-1)^{i+1}.$$

The rogue divisor conjecture. The least prime divisor of $k$ is smaller than $p$:

$$\min\{q\in\mathbb{P}:q\mid k\}<p.$$

Consequently, the preceding upper bound for $|C_i[p]|$ holds with equality. The conjecture would also remove the non-emptiness assumption on the relevant rogue-prime set, but it is not proved in the source.