The revised Newelski conjecture on Hausdorffness of the tau-topology

Let $M$ be a model of an NIP theory, let $G$ be a $0$-definable group in $M$, and let $N\succ M$ be an $|M|^{+}$-saturated elementary extension. Let $S_G^{\operatorname{fs}}(N,M)$ be the semigroup of complete types over $N$ concentrated on $G$ and finitely satisfiable in $M$. Let $\mathcal{M}$ be a minimal left ideal of this semigroup, and let $u\in\mathcal{M}$ be an idempotent. Revised Newelski conjecture. The $\tau$-topology on $u\mathcal{M}$ is Hausdorff. This conjecture is the revised form of Newelski's Ellis group conjecture and predicts Hausdorffness of the ideal group in the NIP setting; the paper proves it for countable NIP groups, while the general case remains open.