Residual Artin conjecture for primitive roots in arithmetic progressions

Let $m$ be an integer that is neither a square nor $-1$, and let $a,b\in\mathbb{Z}$ satisfy $\gcd(a,b)=1$. Residual Artin's conjecture. If the arithmetic progression $a\mathbb{Z}+b$ contains infinitely many primes $q$ for which $m$ is not a quadratic residue modulo $q$, then it contains infinitely many primes $q$ for which $m$ is primitive modulo $q$. This is presented as a strengthening of Artin's conjecture and Dirichlet's theorem, and is used conditionally for the paper's generating-set results.