Compatibly split subvariety characterization for the Grassmannian

Let $\mathrm{Gr}(k,n)$ carry its standard Frobenius splitting. An irreducible subvariety is compatibly split if it is compatibly preserved by this splitting.

Compatibly split positroid conjecture. The only compatibly split irreducible subvarieties of $\mathrm{Gr}(k,n)$ are the closed positroid varieties.

The source states this as a conjecture-like assertion after noting a finiteness theorem for compatibly split subvarieties. The parser supplies no resolution status, and the source passage itself does not establish the characterization.