Caldwell and Gallot's conjecture on the expected number of primorial primes

Let $p_n$ be the $n$-th prime and define the primorial by

$$p_n\#=\prod_{i=1}^{n}p_i.$$

A primorial prime is a prime of the form $p_n\#\pm1$. For a fixed prime bound $p_N$, consider primorial primes with indices $p_n\leq p_N$.

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

$$e^{\gamma}\log p_N.$$

This is motivated by a heuristic using the Mertens product and the ordinary prime probability. It concerns expected counts rather than a proven asymptotic result.