Pillay–Yao conjecture on almost periodicity of global f-generic types

Let $G$ be a group definable in a NIP structure. Say that $G$ is dfg if it has a definable global f-generic type, and call a global f-generic type almost periodic when its orbit under $G$ lies in a minimal closed invariant subset of the relevant type space. Pillay–Yao conjecture. If $G$ is a dfg group definable in a NIP structure, then every global f-generic type of $G$ is almost periodic.

The conjecture concerns the relationship between definability and dynamical behavior of generic types in NIP structures. The paper proves it in the $p$-adic setting using its structure theorem for dfg groups and a result on trigonalizable groups, while the stated general NIP version is not resolved here.