The weak Virtual Surjections conjecture

Let $k\geq 2$, let $\Gamma_1,\dots,\Gamma_n$ be groups of type $\operatorname{FP}_k$, and let $P\leq\Gamma_1\times\dots\times\Gamma_n$ be a subgroup of their direct product. Say that $P$ virtually surjects to $k$-tuples of factors if its projection to every product of $k$ distinct factors has finite index. The weak Virtual Surjections conjecture. If $P$ virtually surjects to $k$-tuples of factors, then $P$ is of type $\operatorname{wFP}_k$, meaning that $H_j(P,\mathbb Z)$ is finitely generated for every $j\leq k$. This is proved in the paper as a consequence of the weak $n$-$(n+1)$-$(n+2)$ theorem.