Is the set of primes such that $gcd(p−1,q−1)=2$ of positive density? For example, for $p,q≤10^4$ the answer is approximately $1/2$.

I was wondering if it were possible to use sieve methods and results such as Siegel-Walfisz Theorem to give a good approximation of primes of this form.

The motivation for the question is for understanding the order of elements in the group $(\mathbb{Z}/pq\mathbb{Z})^∗≃(\mathbb{Z}/p\mathbb{Z})^∗×(\mathbb{Z}/q\mathbb{Z})^*$.