Large-rank conjecture for rank-$k$ flats in matroids with no uniform minor

Let $\ell\ge 2$ be an integer, let $M$ be a rank-$r$ matroid, and let $U_{2,\ell+2}$ denote the rank-$2$ uniform matroid on $\ell+2$ elements. Let $q$ be the largest prime power such that $q\le \ell$. Large-rank conjecture. If $r$ is sufficiently large and $M$ has no $U_{2,\ell+2}$-minor, then, for every integer $k\ge 0$,

$$W_k(M)\le \left[ {r} \atop {k}\right]_{q}.$$

This strengthens the paper's main theorem by removing the requirement that the rank be large compared with $k$. The conjecture is presented as open; the preceding discussion indicates that the rank-$3$ obstruction is believed to be sporadic.