Matroids denser than a clique

The growth-rate function for a minor-closed class M of matroids is the function h where, for each non-negative integer r, h(r) is the maximum number of elements of a simple matroid in M with rank at most r. The Growth-rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential, or infinite. Moreover, if the growth-rate function is quadratic, then h(r)≥(r+12), with the lower bound coming from the fact that such classes necessarily contain all graphic matroids. We characterise the classes that satisfy h(r)=(r+12) for all sufficiently large r.