The asymptotic enumeration conjecture for representable matroids

Let $M$ be a matroid with ground set $[n]$. A nonbasis of a rank-$r$ matroid is an $r$-element subset of $[n]$ that is not a basis. For each integer $n$, let $\delta(n)\in\{0,1\}$ be the remainder of $n$ on division by $2$, and define

$$d(n)=\left(\left\lfloor\frac n2\right\rfloor-1\right)\left(\left\lceil\frac n2\right\rceil-1\right)=\frac14(n^2-\delta(n))-n+1.$$

The asymptotic enumeration conjecture. Asymptotically almost all representable matroids on $[n]$ have rank in $\{\lfloor n/2\rfloor,\lceil n/2\rceil\}$ and exactly $d(n)-1$ nonbases. Asymptotically almost all matroids on $[n]$ with rank in this set and exactly $d(n)-1$ nonbases are representable. Furthermore, the number of representable matroids on $[n]$ is

$$(1+\delta(n)+o(1))\binom{\binom{n}{\lfloor n/2\rfloor}}{d(n)-1}.$$

The theorem preceding this conjecture gives an upper bound of $2^{n^3/4}$ for the number of representable matroids, while the conjecture predicts the precise leading asymptotic through the extremal ranks and nonbasis count. The claim remains open in the supplied source.