Mirzakhani's covering conjecture for affine invariant submanifolds

Let $\cM$ be an affine invariant submanifold in a moduli space of abelian differentials. Say that $\cM$ is defined over $\bQ$ when its field of definition is $\bQ$, and call a connected component of a stratum a full stratum component. A translation surface in $\cM$ may cover a quadratic differential, equivalently a half-translation surface, of smaller genus.

Mirzakhani's conjecture. If an affine invariant submanifold $\cM$ is defined over $\bQ$, and $\cM$ is not a connected component of a stratum, then every translation surface in $\cM$ covers a quadratic differential (half-translation surface) of smaller genus.

The conjecture proposes a direct relation between the field of definition and the global structure of affine invariant submanifolds. The supplied text gives no resolution status.