The natural-density conjecture for primitive divisors of n^2+b

Let $b$ be an integer such that $-b$ is not an integer square, and define

$$\rho_b(x)=\left|\{n\le x:n^2+b\text{ has a primitive divisor}\}\right|.$$

Natural-density conjecture. As $x$ tends to infinity,

$$\rho_b(x)\sim x\log 2.$$

This would identify the natural density of indices whose terms $n^2+b$ have a primitive divisor as $\log 2$. The paper has already established fixed positive upper and lower bounds for $\rho_b(x)/x$, but the asserted asymptotic is presented only as a suggestion and remains unresolved in the supplied text.