A matroid is called a paving matroid if all circuits of the matroids are of cardinality $k$ or $k+1$, where $k$ is the rank of the matroid. A matroid $(E,I)$ is called a linear matroid if there is a field $F$ and a matrix $A$ over $F$ such that $E$ contains the columns of $A$ and $I$ contains all independent sets (as in linear algebra) among subsets of $E$.
Question: A matroid that is not a paving Matroid is necessarily a linear matroid?
This question seems to be open, and a positive answer to it implies a long-standing conjecture in the asymptotics of matroids:
Conjecture 1: Almost all matroids are paving.
This is Conjecture 1.6 in On the asymptotic proportion of connected matroids following speculation by Crapo and Rota in the 1970's.
In Almost All Matroids Are Non-representable it is shown that almost all matroids are non-linear. So if every non-paving matroid is linear, then every non-linear matroid is paving. And since almost all matroids are non-linear, it follows that almost all matroids are paving.