4 problems

Caldwell and Gallot's conjecture. The expected number of primorial primes of each of the forms $p_n\#\pm1$ less than or equal to $p_N\#\pm1$, respectively, are both approximately

Universal primorial conjecture. There are expected to be infinitely many numbers $x=Kp_n\#+g$ such that both $x$ and $x+2$ are prime.

Lillie's conjecture. The total expected number of primorial twin prime pairs is approximately three.

Banderier's conjecture. Either $p_n\#-q=1$ or $p_n\#-q$ is a prime.