Generalized Artin density conjecture for polynomial prime values

Let $f(x)\in\mathbb{Z}[x]$ produce infinitely many primes. For an integer $g$, define

$$\delta_g(f,x):=\frac{\#\{0\leq n\leq x;~f(n)=p~\text{is an Artin prime for}~g\}}{\#\{0\leq n\leq x;~f(n)~\text{is prime}\}}.$$

Here an Artin prime for $g$ is a prime for which $g$ is a primitive root modulo that prime. Generalized Artin conjecture. The limit

$$\delta_g(f)=\lim_{x\rightarrow\infty}\delta_g(f,x)$$

exists. This is proposed as a generalization of the classical Artin conjecture; the paper notes that a conjectural explicit value for the density is difficult in general.