The rank-three paving matroid enumeration conjecture

Let $p(n,r)$ denote the number of paving matroids on an $n$-element ground set of rank $r$, and let $p_k(n,r)$ denote the number of paving matroids without hyperplanes of cardinality greater than $r+k$. Write $f(n)\approx g(n)$ when their logarithms are asymptotically equivalent. Rank-three paving matroid enumeration conjecture. There is a constant $c>-2$ such that, for $r=3$,

$$\ln p(n,r)\approx \ln p_1(n,r)=\frac{1}{n-r+1}\binom{n}{r}\left(\ln(n-r+1)+c+o(1)\right)\quad\text{as }n\to\infty.$$

The conjecture proposes that, in rank three, paving matroids whose hyperplanes have cardinality at most $r+1$ already determine the asymptotic logarithmic count of all paving matroids. It would identify the dominant contribution from matroids with no hyperplanes larger than $4$, and explains the remaining gap between the known upper and lower bounds.