Kochloukova–Lima homological Virtual Surjection Conjecture

Let $2\leqslant k\leqslant m$ be integers. For groups $G_{1},\ldots,G_{m}$, a subgroup $P\leqslant G_{1}\times\cdots\times G_{m}$ virtually surjects on $k$ factors if, for every $1\leqslant i_{1}<\cdots<i_{k}\leqslant m$, its image in $G_{i_{1}}\times\cdots\times G_{i_{k}}$ has finite index. Homological Virtual Surjection Conjecture. If each $G_i$ is of type $\operatorname{FP}_k$ and $P$ virtually surjects on $k$ factors, then $P$ is of type $\operatorname{FP}_k$. The paper proves this conjecture for discrete groups, while the source records earlier results for $k=2$.