The nilpotent-by-abelian FP_m conjecture

Let $G$ be a finitely generated nilpotent-by-abelian group. Let $N$ be a nilpotent normal subgroup with $Q=G/N$ abelian, and consider the factors of the lower central series of $N$. Suppose that the direct sum of all these factors is a finitely generated $\mathbb{Z}(G/N)$-module and is $m$-tame. The nilpotent-by-abelian $\operatorname{FP}_m$ conjecture. Then $G$ is of type $\operatorname{FP}_m$. The conjecture is motivated by the centraliser theorem, its corollaries, and Groves' sufficient conditions; the paper notes that the corresponding Bredon conjecture holds for $m=2$, but leaves this general finiteness assertion open.