Mirzakhani's arithmetic affine-manifold conjecture

Let $\mathcal{M}$ be an affine manifold in a stratum of translation surfaces. Its rank is the dimension of its tangent space projected to absolute cohomology divided by $2$, and $k(\mathcal{M})$ denotes its field of affine definition. Mirzakhani's conjecture. If

$$\operatorname{rank}(\mathcal{M}) \geq 2 \quad\text{and}\quad k(\mathcal{M})=\mathbb{Q},$$

then $\mathcal{M}$ is either the entire stratum or a covering of a stratum of Abelian or quadratic differentials. The conjecture generalizes the known rank-one description of arithmetic affine manifolds as coverings of strata of marked tori; the paper confirms it for the rank-two cases in genus three with two zeros.