The sparse paving minor conjecture

A matroid $N$ is sparse paving if every dependent set of cardinality equal to its rank is a circuit-hyperplane. A matroid $M$ has $N$ as a minor if $N$ is obtained from $M$ by a sequence of deletions and contractions.

Sparse paving minor conjecture. Let $N$ be a fixed sparse paving matroid. Asymptotically almost every matroid has an $N$-minor.

The conjecture is known when $N$ is any uniform matroid, but the paper says it remains open even for small examples such as $W^3$ and $M(K_4)$. It asks whether every fixed sparse paving matroid occurs as a minor of almost every matroid.