$r(M)$-subsets of a 3-connected matroid $M$

It is proved in Lowrance, Oxley, Semple, and Welsh - On properties of almost all matroids that almost all matroids are 3-connected asymptotically. Also, it is conjectured that almost all matroids are sparse paving. The latter conjecture is proved logarithmically in Pendavingh and van der Pol - On the number of bases of almost all matroids. There is a characterization for $r(M)$-subsets of a sparse paving matroid $M$ that is every $r(M)$-subset is either a basis or very close to being a basis (circuit- hyperplane). Since both classes of 3-connected matroids and sparse paving matroids are very large families of matroids I have the impression that they must have a large intersection, and probably one may claim that for a 3-connected matroid $M$, most of $r(M)$-subsets are kind of close to being a basis, if not a basis. Is there any result or evidence supporting this?