Mirzakhani's dimension conjecture for non-rational affine invariant submanifolds

Let $\cM$ be an affine invariant submanifold, let $p(T(\cM))$ denote the projection of its tangent bundle to the absolute cohomology bundle, and let $\bQ$ be the field of rational numbers. An affine invariant submanifold is defined over $\bQ$ when its field of definition is $\bQ$; the dimension below is the complex dimension.

Mirzakhani's conjecture. If an affine invariant submanifold $\cM$ is not defined over $\bQ$, then $p(T(\cM))$ has dimension $2$.

This conjecture concerns the connection between the field of definition and the global structure of affine invariant submanifolds. The supplied text gives no resolution status.