Newelski's Ellis group conjecture for definable flows

Let $M$ be a first-order structure and let $G$ be an $M$-definable group. The space $S_{ext,G}(M)$ is the universal definable flow of $G$ over $M$, and $G^{00}$ is the smallest type-definable subgroup of $G$ of bounded index. Under relatively tame assumptions, generally understood to include $NIP$, Newelski's Ellis group conjecture. The Ellis group of $S_{ext,G}(M)$ is isomorphic to

$$G/G^{00}.$$

The source identifies this as Newelski's conjecture and separately records positive results in several tame settings, while the broader conjecture has counterexamples. This environment is a restatement of the earlier prose formulation and is therefore merged into the same database row.