The structural conjecture for extremal matroids in exponentially dense classes

Let $q$ be a prime power, and let $\mathcal{M}$ be a base-$q$ exponentially dense minor-closed class of matroids, meaning that its growth rate is eventually of order $q^n$. For a non-negative integer $k$, a $k$-element projection of a matroid $N$ is a matroid of the form $N/C$, where $N\setminus C=M$ and $C$ is a $k$-element set of $N$. Structural conjecture. There exists an integer $k \ge 0$ such that, if $M \in \mathcal{M}$ is a simple matroid of sufficiently large rank with

$$|M|=h_{\mathcal{M}}(r(M)),$$

then $M$ is the simplification of a $k$-element projection of a projective geometry over $\operatorname{GF}(q)$. This predicts the exact structure of sufficiently large extremal matroids, beyond the conjectured classification of their eventual growth rates.