Weakly generic and almost periodic types in definably amenable groups

Let $G$ be a definable, definably amenable group defined over either an $o$-minimal structure or a $p$-adically closed field. Say that $G$ is stationary when it has the stationarity property considered in the paper, and let weakly generic and almost periodic types be understood in the relevant type spaces. The weakly generic–almost periodic conjecture. If $G$ is not $\mathrm{dfg}$, then the set of weakly generic types coincides with the set of almost periodic types if and only if $G$ is stationary. The preceding counterexamples show that non-stationarity can cause these sets to differ, but the converse in this general setting remains open.