A group $G$ is of type $FP_2$ if it admits a partial projective resolution of $\mathbb{Z}G$-modules $$ P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb{Z} \rightarrow 0$$ with each $P_i$ being finitely generated and projective.

In the event $G$ acts freely and cocompactly on a space $X$ with trivial first homology, then $G$ is of type $FP_2$. The converse holds if $G$ has cohomological dimension $2$, but does it hold in general?