Is the set of prime pairs such that $gcd(p-1,q-1) = 2$ of positive density? For example, for $p,q \leq 10^4$ the answer is approximately $1/2$.
I was wondering if it were possible to use sieve methods and results such as the Siegel-Walfisz Theorem to give a good approximation of prime pairs of this form.
The motivation for the question is for understanding the order of elements in the group $(\mathbb{Z}/pq\mathbb{Z})^*\simeq (\mathbb{Z}/p\mathbb{Z})^*\times (\mathbb{Z}/q\mathbb{Z})^*$.