Bateman–Selfridge–Wagstaff conjecture on Mersenne-prime forms

Let $n$ be an odd positive integer, and let $M(n)=2^n-1$. Consider the three statements

$$\begin{aligned} \text{(a)}&\quad n=2^k\pm 1\ \text{or}\ n=4^k\pm 1\ \text{for some }k>1,\\ \text{(b)}&\quad M(n)\text{ is prime},\\ \text{(c)}&\quad \frac{2^n+1}{3}\text{ is prime}. \end{aligned}$$

Bateman–Selfridge–Wagstaff conjecture. If two of these three statements are true, then the third is also true.

The source attributes this conjecture to Bateman, Selfridge, and Wagstaff and places it among proposed characterizations of when Mersenne numbers are prime. Its resolution status is not specified in the supplied text.