The sparse paving minor conjecture for almost all sparse paving matroids

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

This is presented as a harder variant of the conjecture that almost every matroid has a fixed sparse paving minor. The paper states that it remains open, even though the broader sparse paving conjecture would identify sparse paving matroids as almost all matroids.