Small, $nm$-stable compact $G$-groups

We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$ is nilpotent-by-finite, and if additionally $\text{NM}(H)\le \omega$, then $H$ is abelian-by-finite. Both results are significant steps towards the proof of the conjecture that each small, $nm$-stable compact $G$-group is abelian-by-finite. We give examples of small, $nm$-stable compact $G$-groups of infinite ordinal $\text{NM}$-rank, providing counter-examples to the $\text{NM}$-gap conjecture.