The finite-Krull-dimension conjecture for noetherian group algebras

Let $k$ be a field and let $G$ be a group such that $kG$ is noetherian and has finite right Krull dimension $\operatorname{Kdim} kG$. Let $\mathcal{P}$ denote the class of polycyclic-by-finite groups.

Finite-Krull-dimension conjecture. The following assertions hold:

1. $G\in\mathcal{P}$.
2. If $G$ is a torsion group, then $G$ is finite.

This is presented as a potentially easier special case of the broader noetherian group-algebra question. The source explicitly says that both the general assertion and its torsion restriction remain open.