Are pseudo-Anosov foliations dense?

The pseudo-Anosov foliations form a subset of $\mathcal{PMF}(F)$ which is invariant under the action of the mapping class group $MCG(F)$, because if $\Lambda_+(\phi) \in \mathcal{PMF}(F)$ is the stable lamination of a pseudo-Anosov $\phi \in MCG(F)$ then $\psi(\Lambda_+(\phi)) = \Lambda_+(\psi\phi\psi^{-1})$ is the stable lamination of the pseudo-Anosov $\psi\phi\psi^{-1}$.

So the fact that the pseudo-Anosov foliations are dense is an immediate corollary of Theorem 6.1 from "Thurston's Work on Surfaces" (originally published in 1979 as "Travaux de Thurston sur les Surfaces"):

The action of $MCG(F)$ on $\mathcal{PMF}(F)$ is minimal, meaning that every orbit is dense.