Asymptotic counting conjecture for q-matroids of rank r on a 2r-dimensional space

Let $s_q(r,2r)$, $p_q(r,2r)$, and $m_q(r,2r)$ denote respectively the numbers of sparse paving, paving, and all $q$-matroids of rank $r$ on a $2r$-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 each prime power $q$, we have

$$q^{r(r-1)}\approx\log(s_q(r,2r))\approx\log(p_q(r,2r))\approx\log(m_q(r,2r))$$

as $r\to\infty$. This would extend the corresponding asymptotic equality for classical matroids to $q$-matroids and would show that sparse paving, paving, and all $q$-matroids have the same logarithmic asymptotic order in the middle-rank regime.