Petrykowski's bounded-orbit conjecture for definable groups

Let $G$ be a definable group. Say that $G$ is definably amenable if it admits a left $G(M)$-invariant Keisler measure on $G$ over a model $M$. For a sufficiently saturated model $\bar M$ and a type $p(x)\in S_G(\bar M)$, say that $p$ has bounded orbit if its orbit under the left action of $G(\bar M)$ has cardinality smaller than the saturation cardinal; say that $G$ has a bounded orbit if some such type does.

Petrykowski's conjecture. If $G$ has a bounded orbit, then $G$ is definably amenable.

The conjecture connects the existence of a small orbit on the type space with invariant-measure amenability for definable groups. The source attributes it to Petrykowski through Newelski; its resolution is not specified in the supplied text.