Mirzakhani's conjecture on higher-rank orbit closures of translation surfaces

Let an orbit closure be an orbit closure of the $SL(2,\bR)$ action on a stratum of translation surfaces, and let its rank be the rank associated with the corresponding affine invariant submanifold. Mirzakhani's conjecture. Every orbit closure of rank at least $2$ is trivial. This conjecture concerns the classification of $SL(2,\bR)$-orbit closures and seeks to explain why so few higher-rank examples are known; the supplied text does not state whether it has been resolved.