Round matroid hyperplane bound conjecture

Let $\ell\geq 2$ be an integer, and let $M$ be a matroid. A round matroid is one whose ground set is not the union of two hyperplanes, equivalently one that is vertically $k$-connected for all $k$. A $U_{2,\ell+2}$-minor is a minor isomorphic to the rank-$2$ uniform matroid on $\ell+2$ elements; $r(M)$ denotes the rank of $M$.

Round matroid hyperplane bound conjecture. If $M$ is round, has sufficiently large rank, and has no $U_{2,\ell+2}$-minor, then $M$ has at most

$$\tfrac{\ell^{r(M)-1}}{\ell-1}$$

hyperplanes.

This is proposed as a possible correct upper bound under very high rank and connectivity after the general Bonin bound is refuted. The source presents it cautiously, and no resolution is supplied.