Let $\mathcal{P}$ be a set of prime numbers of relative density $\kappa \in (0, 1)$, which means that $$\#\left(\mathcal{P} \cap [1, x]\right) = \kappa \,\pi(x) + E(x) \quad (x \to \infty)$$ for a "suitable" error term $E(x)$ (of course, $E(x) = o(\pi(x))$).

Let $\mathcal{S}$ be the set of natural numbers having no prime factor in $\mathcal{P}$.

Question: Do we have an asymptotic formula for $\#\left(\mathcal{S} \cap [1, x]\right)$ ?

For $\kappa \in (0, 1/2)$, Corollary 1.2 of [1] gives that $$\#\left(\mathcal{S} \cap [1, x]\right) \sim C(\mathcal{P}) \frac{x}{(\log x)^\kappa} \quad (x \to +\infty)$$ where $C(\mathcal{P})$ is a positive constant depending on $\mathcal{P}$. I guess that something similar holds in the whole range $\kappa \in (0, 1)$, but I could not find such a result. Thank you for any help.

[1] Iwaniec and Kowalski, Analytic Number Theory (2004)