Density of mild mixing property for vertical flows of Abelian differentials

We prove that if $g\geq 2$ then the set of all Abelian differentials $(M,\omega)$ for which the vertical flow is mildly mixing is dense in every stratum of the moduli space $\mathcal{H}_g$. The proof is based on a sufficient condition for special flows over irrational rotations and under piecewise constant roof functions to be mildly mixing.