Classification of Turán basis densities for minor-closed classes

Determine whether the classification of extremal growth-rate functions for minor-closed classes of matroids yields a corresponding classification of their Turán basis densities pi_M(r).

Background

The Growth Rate Theorem states that the growth-rate function of a minor-closed matroid class has one of four broad behaviors: linear, quadratic, exponential, or infinite. The paper asks whether these structural growth-rate categories have an analogous consequence for the asymptotic density of bases.

This is posed as a general open-ended question extending the paper’s specific extremal problems beyond classes defined only by excluding a uniform minor.