  We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$
is nilpotent-by-finite, and if additionally $\NM(H) \leq \omega$, 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
$\NM$-rank, providing counter-examples to the $\NM$-gap conjecture.
