Lillie's conjecture on primorial twin prime pairs

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

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

A primorial twin prime pair is a pair $(p_n\#-1,p_n\#+1)$ in which both numbers are prime.

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

The paper gives evidence that $n=2,3,5$ yield the three such pairs, namely the instances suggested by the heuristic estimate.