Conjecture on properly coherent group rings of amenable groups

Let $G$ be a finitely generated amenable group. Its integral group ring is denoted by $\mathbb{Z}G$. A group ring is properly coherent when it is coherent but not Noetherian. A properly ascending HNN extension is an ascending HNN extension whose associated embedding is proper; here the vertex group is required to be virtually polycyclic. Conjecture on properly coherent group rings. $\mathbb{Z}G$ is properly coherent if and only if $G$ is a properly ascending HNN extension with virtually polycyclic vertex group.

This is presented as an analogue of Baer's conjecture for finitely generated amenable groups. It relates proper coherence of the integral group ring to the structure of the group as a properly ascending HNN extension, and remains open in the source.