The question is in the title. Fix a field $k$. Let $P_n$ be the poset of proper nonempty affine subspaces of $k^n$ under inclusion. The geometric realization $|P_n|$ is $n$-dimensional. Is it $(n-1)$-connected? Or at least highly connected?

If I removed the word “affine” and thus required the subspaces to pass through the origin, this would be the usual Tits building, which is $(n-1)$-dimensional and by the Solomon-Tits theorem is $(n-2)$-connected.