Matroids with no $U_{2,n}$-minor and many hyperplanes

We construct, for every $r\ge 3$ and every prime power $q>10$, a rank-$r$ matroid with no $U_{2,q+2}$-minor, having more hyperplanes than the rank-$r$ projective geometry over $\mathrm{G}\mathrm{F}(q)$.