Geelen's Growth Rate Conjecture for minor-closed matroid classes

For a matroid $M$ and a positive integer $a$, let $\tau_a(M)$ denote the minimum number of sets of rank at most $a$ needed to cover $E(M)$, and let $r(M)$ denote the rank of $M$. A class of matroids is minor-closed if it contains every minor of each of its members. Let $\mathcal{M}$ be a minor-closed class of matroids.

Growth Rate Conjecture. For every integer $a ge 1$, there is an integer $c>0$ such that at least one of the following holds:

$$\tau_a(M) \le c r(M)$$

for all $M\in\mathcal{M}$;

$$\tau_a(M) \le c r(M)^2$$

for all $M\in\mathcal{M}$ and $\mathcal{M}$ contains all graphic matroids or all bicircular matroids; there is a prime power $q$ such that

$$\tau_a(M) \le c q^{r(M)}$$

for all $M\in\mathcal{M}$ and $\mathcal{M}$ contains all $\operatorname{GF}(q)$-representable matroids; or $\mathcal{M}$ contains all rank-$(a+1)$ uniform matroids. This conjecture refines the polynomial-exponential growth-rate theorem: the theorem in the paper establishes a weaker polynomial bound in the first alternative, while the linear and quadratic alternatives, and the complete classification, remain the conjectural part.