The K(pi,1) conjecture for connected components of strata of holomorphic 1-forms

Let ${\cal M}_{\bf d}$ be a moduli space of holomorphic 1-forms with prescribed zero orders, and let ${\cal M}$ be one of its connected components. A group is commensurable with another group if they have isomorphic finite-index subgroups. The K(pi,1) conjecture. Each connected component ${\cal M}$ of ${\cal M}_{\bf d}$ has homotopy type $K(\pi,1)$, where $\pi$ is a group commensurable with some mapping class group. The classification of connected components is known, including hyperelliptic components and the spin-parity components, but the homotopy type asserted here is not established in the source.