Asymptotic counting conjecture for q-matroids with fixed rank

Let $s_q(r,n)$, $p_q(r,n)$, and $m_q(r,n)$ denote respectively the numbers of sparse paving, paving, and all $q$-matroids of rank $r$ on an $n$-dimensional space over the finite field with $q$ elements, where $q$ is a prime power. For nonnegative functions, write $f(t)\approx g(t)$ as $t\to\infty$ when

$$\lim_{t\to\infty}\frac{f(t)}{g(t)}\in\mathbb{R}_{>0}.$$

Asymptotic counting conjecture. For all $r\geq 2$ and prime powers $q$, we have

$$\log(s_q(r,n))\approx\log(p_q(r,n))\approx\log(m_q(r,n))\approx q^{(n-r)(r-1)}(n-r)$$

as $n\to\infty$. The conjecture predicts that the number of sparse paving $q$-matroids is already asymptotically as large as the numbers of paving and all $q$-matroids in this fixed-rank regime, closing the gap between the current lower and upper bounds.