When the sieve works II

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matom\"aki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not too many large primes are sieved out in the sense that $$\sum_{\substack{p \in \mathcal{P} \\ x^{1/v} < p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1 + \varepsilon}{u},$$ for some $\varepsilon > 0$ and $v \geq u \geq 1$, then $$\Psi(x, \mathcal{P}) \gg_{\varepsilon, v} x \prod_{\substack{p \leq x\\ p \notin\mathcal{P}}} \left(1 - \frac{1}{p}\right).$$