Pillay–Yıldırım conjecture on weakly generic and almost periodic types

Let $T$ be an $NIP$ theory, let $M$ be a model of $T$, and let $G$ be a definably amenable group definable in $M$. Write $S_G(M)$ for the space of types over $M$ concentrating on $G$; a type is weakly generic or almost periodic in the sense of the corresponding notions for the $G$-flow. Assume that $G$ has a global definably weakly generic type. Pillay–Yıldırım's conjecture. For every $p otin S_G(M)$, $p$ is weakly generic if and only if it is almost periodic. This conjecture asks whether the two classes coincide under the stated hypothesis; the paper proves it for trigonalizable algebraic groups over ${\mathbb Q}_p$, but the general $NIP$ case remains open.