Determine existence of matroid with some barrier given

Question

Let $E$ be a finite ground set. Let $\mathcal{L}$ (as lower barrier) and $\mathcal{U}$ (upper) be subsets of $2^E$. How can we determine whether there is some matroid $\mathcal{M}=(E,\mathcal{I})$ with $\mathcal{L}\subseteq\mathcal{I}\subseteq 2^E\setminus\mathcal{U}$? Is there any result of good characterization? If it is, how fast can we determine it (with some oracles of $\mathcal{L}$ and $\mathcal{U}$)? Thank you for your attention.