The Growth Rate Conjecture for minor-closed classes of matroids

Let $a\geq 1$ be an integer, and let $\mathcal{M}$ be a minor-closed class of matroids. For a matroid $M$, write $r(M)$ for its rank and let $\tau_a(M)$ denote the minimum number of rank-at-most-$a$ sets whose union is the ground set of $M$. Growth Rate Conjecture. One of the following holds:

1. $\tau_a(M)\leq c_{\mathcal{M}}r(M)$ for all $M\in\mathcal{M}$;
2. $\tau_a(M)\leq c_{\mathcal{M}}r(M)^2$ for all $M\in\mathcal{M}$, and $\mathcal{M}$ contains all graphic matroids or all bicircular matroids;
3. there is a prime power $q$ such that $\tau_a(M)\leq c_{\mathcal{M}}q^{r(M)}$ for all $M\in\mathcal{M}$, and $\mathcal{M}$ contains all $\operatorname{GF}(q)$-representable matroids; or
4. $\mathcal{M}$ contains all rank-$(a+1)$ uniform matroids.

Here $c_{\mathcal{M}}$ is a constant depending only on $\mathcal{M}$. The conjecture was proved by Geelen, Kabell, Kung and Whittle, and was stated in that work as the Growth Rate Theorem.