Kuckuck's $n$-$(n+1)$-$(n+2)$ conjecture for group finiteness properties

Let $n$ be a nonnegative integer, and let

$$1\to N_{1}\to G_{1}\to Q\to1,\qquad 1\to N_{2}\to G_{2}\to Q\to1$$

be short exact sequences of groups. For epimorphisms $G_1\to Q$ and $G_2\to Q$, their fibre product is

$$G_{1}\times_{Q}G_{2}=\{(g_{1},g_{2})\in G_{1}\times G_{2}\mid \phi_{1}(g_{1})=\phi_{2}(g_{2})\}.$$

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname F_n$, $G_1$ and $G_2$ are of type $\operatorname F_{n+1}$, and $Q$ is of type $\operatorname F_{n+2}$, then $G_{1}\times_QG_{2}$ is of type $\operatorname F_{n+1}$. The cases $n=0$ and $n=1$ were known from the cited results, and the paper establishes the conjecture in general; it is also used to deduce the Virtual Surjection Conjecture.