The expanding-map lifting conjecture for infra-nilmanifolds

Let $G$ be a connected, simply connected nilpotent Lie group, let $\Gamma$ be a discrete cocompact subgroup of the affine group defining the infra-nilmanifold $\Gamma \backslash G$, and let $N$ be the Fitting subgroup of $\Gamma$, so that $N \backslash G$ is a finitely covering nilmanifold.

The expanding-map lifting conjecture. The infra-nilmanifold $\Gamma \backslash G$ admits an expanding map if and only if the nilmanifold $N \backslash G$ admits an expanding map.

The forward implication follows by lifting an expanding map to the finite nilmanifold cover, while the converse is the substantive assertion. The conjecture is known for Lie algebras of homogeneous type, but the source gives no general resolution.