Non-coherence conjecture for poly-free and poly-surface groups

Let $G$ be a poly-free group or a poly-surface group of length $n$ with nonvanishing Euler characteristic. For an integer $m$, the group $G$ is $m$-coherent if every finitely generated subgroup of $G$ has finitely generated homology in degrees at most $m$.

Non-coherence conjecture. For all $0 \leq m \leq n-1$, $G$ is not $m$-coherent.

This conjecture generalizes the preceding obstruction results for algebraically fibered poly-free groups. It predicts that the excessive-homology assumption used there is unnecessary for poly-free and poly-surface groups, while the source indicates that proving the statement could be challenging.