  We obtain a new weak Hilton-Milner type result for intersecting families of
$k$-spaces in $\mathbb{F}_q^{2k}$, which improves several known results. In
particular the chromatic number of the $q$-Kneser graph $qK_{n:k}$ was
previously known for $n > 2k$ (except for $n=2k+1$ and $q=2$) or $k < q \log q
- q$. Our result determines the chromatic number of $qK_{2k:k}$ for $q \geq 5$,
so that the only remaining open cases are $(n, k) = (2k, k)$ with $q \in \{ 2,
3, 4 \}$ and $(n, k) = (2k+1, k)$ with $q = 2$.
