The $n$-$(n+1)$-$(n+2)$ conjecture for fibre products

Let

$$N_1\hookrightarrow\Gamma_1\xrightarrow{\pi_1}Q,\qquad N_2\hookrightarrow\Gamma_2\xrightarrow{\pi_2}Q$$

be short exact sequences of groups, and define their fibre product by

$$P=\{(\gamma_1,\gamma_2)\in\Gamma_1\times\Gamma_2\mid \pi_1(\gamma_1)=\pi_2(\gamma_2)\}.$$

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_1$ is of type $\operatorname{F}_n$, both $\Gamma_1$ and $\Gamma_2$ are of type $\operatorname{F}_{n+1}$, and $Q$ is of type $\operatorname{F}_{n+2}$, then $P$ is of type $\operatorname{F}_{n+1}$. The case $n=1$ is the proved $1$-$2$-$3$ Theorem; the general statement remains open.