A Density Hales-Jewett Theorem for matroids

We show that, if $\alpha > 0$ is a real number, $n \ge 2$ and $\ell \ge 2$ are integers, and $q$ is a prime power, then every simple matroid $M$ of sufficiently large rank, with no $U_{2,\ell}$-minor, no rank-$n$ projective geometry minor over a larger field than $\GF(q)$, and satisfying $|M| \ge \alpha q^{r(M)}$, has a rank-$n$ affine geometry restriction over $\GF(q)$. This result can be viewed as an analogue of the Multidimensional Density Hales-Jewett Theorem for matroids.