Elementary split matroid asymptotic capture conjecture

For each positive integer $n$, let $\operatorname{sp}(n)$ and $\operatorname{es}(n)$ denote the numbers of sparse paving and elementary split matroids on $[n]$. Let $\operatorname{sp}'(n)$ and $\operatorname{es}'(n)$ denote the corresponding numbers up to isomorphism. Elementary split capture conjecture.

$$\lim_{n\to\infty}\frac{\operatorname{es}(n)-\operatorname{sp}(n)}{\operatorname{mat}(n)-\operatorname{sp}(n)}=1\qquad\text{and}\qquad\lim_{n\to\infty}\frac{\operatorname{es}'(n)-\operatorname{sp}'(n)}{\operatorname{mat}'(n)-\operatorname{sp}'(n)}=1.$$

The conjecture proposes that elementary split matroids asymptotically account for the non-sparse-paving matroids, complementing the sparse-paving predominance conjecture. Its resolution is not given.