Baer's conjecture on Noetherian integral group rings

Let $G$ be a group. Its integral group ring is denoted by $\mathbb{Z}G$. Baer's conjecture. $\mathbb{Z}G$ is Noetherian if and only if $G$ is virtually polycyclic.

The forward implication is the open part attributed to Baer, while the converse was proved by Philip Hall in the 1950s. The result is known for elementary amenable groups, and the conjecture can be restricted to amenable groups because a group with Noetherian integral group ring must be amenable.