Nonarithmetic rank-one sparsity conjecture for orbit closures

A rank-one orbit closure is an affine invariant submanifold whose rank is one. It is nonarithmetic when it is not arithmetic. A sequence of orbit closures equidistributes in a component of a stratum when its distribution becomes uniform there in the sense of Teichmüller dynamics.

Nonarithmetic rank-one sparsity conjecture. Outside of genus two, a sequence of nonarithmetic rank-one orbit closures cannot equidistribute in a component of a stratum.

The conjecture is presented as a consequence of the Mirzakhani conjectures and would imply finiteness alternatives for geometrically primitive Teichmüller curves. The paper proves the relevant higher-dimensional hyperelliptic results, but the supplied text does not state a general resolution of this formulation.