The non-trivial self-cover conjecture for infra-nilmanifolds

Let $G$ be a connected, simply connected nilpotent Lie group, let $\Gamma$ be an almost-Bieberbach group defining the infra-nilmanifold $\Gamma \backslash G$, and let $N$ be the Fitting subgroup of $\Gamma$. The quotient $N \backslash G$ is the associated nilmanifold, and a group is co-Hopfian if every injective endomorphism is an automorphism.

The non-trivial self-cover conjecture. The infra-nilmanifold $\Gamma \backslash G$ admits a non-trivial self-cover if and only if the nilmanifold $N \backslash G$ admits an expanding map. Equivalently, the almost-Bieberbach group $\Gamma$ is co-Hopfian if and only if its Fitting subgroup $N$ is co-Hopfian.

The conjecture asserts that all obstructions to non-trivial self-covers of infra-nilmanifolds already occur for their nilmanifold covers. The source gives no general resolution status.