Multicomponent EMM conjecture

Let $\mathcal M$ be a closed $GL(2,\mathbb R)$-invariant subset of a stratum of multicomponent surfaces, and assume $\mathcal M$ consists entirely of surfaces all of whose components have equal area. Then $\mathcal M$ is equal to the set of surfaces all of whose components have equal area in a finite union of affine invariant submanifolds. Furthermore:

1. There are only countably many closed $GL(2,\mathbb R)$-invariant subsets of the locus where all components have equal area.
2. For any irreducible affine invariant submanifold, locally the absolute periods of any component determine those of all other components. In particular, the ratio of areas of the components is constant.
3. The projection of any affine invariant submanifold to any subset of the set of components is equal to the complement of a finite union of affine invariant submanifolds in a larger affine invariant submanifold.
4. The bundle $H^1$ over any affine invariant submanifold is semisimple.

This is a multicomponent extension of the Eskin–Mirzakhani–Mohammadi classification picture for affine invariant sets. The source presents these assertions as a conjectural description of equal-area multicomponent orbit closures; their resolution is not established by the supplied context.