Artin's conjecture for the prime base 2

For a prime $p$, let $\mathbb{Z}_p^\times$ denote the multiplicative group of nonzero elements of $\mathbb{Z}_p$.

Artin's conjecture. There are infinitely many primes $p$ such that $2$ is a generator of the multiplicative group $\mathbb{Z}_p^\times$.

This is the classical Artin primitive-root conjecture, cited by the source as an approach to finding primes $p$ with $\operatorname{aw}(\mathbb{Z}_p,3)=3$. The existence of an infinite family of such primes remains open.