The Bieri–Stallings connectivity-at-infinity conjecture

For each integer $n\geq 2$, let $B_n$ be the kernel of the homomorphism

$$F_2\times\cdots\times F_2\longrightarrow\mathbb Z$$

from the $n$-fold direct product of $F_2$ sending every generator to $1$.

The Bieri–Stallings connectivity-at-infinity conjecture. For $n\geq 2$, the group $B_n$ is $(n-2)$-connected at infinity.

The source notes that known results imply $(n-3)$-connectedness for $n\geq 4$, while the cases $B_2$ and $B_3$ provide evidence for the conjecture. The full assertion remains unresolved in the supplied material.