Bonin's hyperplane bound for matroids with no uniform-minor

Let $q$ be a prime power, and let $M$ be a matroid of rank $r$ with no $U_{2,q+2}$-minor. A hyperplane of $M$ is a rank-$r-1$ flat of $M$.

Bonin's conjecture. The matroid $M$ has at most

$$\tfrac{q^r-1}{q-1}$$

hyperplanes.

This is a special case of a conjecture due to Bonin. The paper's abstract states that, for every $r\geq 3$ and every prime power $q>10$, there are rank-$r$ matroids with no $U_{2,q+2}$-minor having more hyperplanes than the rank-$r$ projective geometry over $\operatorname{GF}(q)$, so the stated bound is refuted in that range.