Real-representable matroid density conjecture

Let $ll,t in N$ with $ll ge 3$, and let $\falM$ be the class of $R$-representable matroids with no $U_{2,ll+2}$-minor and no $M(K_t)$-minor. Here $\alpha$ is the constant appearing in the corresponding graph-theoretic extremal bound. Real-representable density conjecture.

$$h_{\falM}(n)=2(\alpha+o_{t,ll}(1))t\sqrt{\log t}\cdot n.$$

The factor of two reflects the largest multiplicative subgroup of the real numbers. This is presented as a conjecture for characteristic-zero representations and is open.