Fundamental groups of strata of abelian differentials of low codimension

We show that for a stratum of projectivized abelian differentials with sufficiently many simple zeroes, the inclusion into the appropriate moduli space of pointed curves induces an injection at the level of orbifold fundamental group, thereby confirming part of a 1997 conjecture of Kontsevich-Zorich. Combined with previous work of the author with Calderon, this describes the orbifold fundamental groups of these strata as framed mapping class groups. Our approach is algebro-geometric in nature, and is based on Shimada's techniques for computing fundamental groups via morphisms of varieties that behave like fiber bundles away from loci of high codimension.