Asymptotic uniformity conjecture for $(k,l)$-uniform matroids

Let $m_n(k,l)$ be the number of all $(k,l)$-uniform matroids on the set $[n]$. Asymptotic uniformity conjecture. If $(k,l)\neq (1,1)$, then either

$$\lim_{n\to\infty} \frac{m_n(k,l)}{m_n(k+1,l)}\to 1;$$

or, dually,

$$\lim_{n\to\infty} \frac{m_n(k,l)}{m_n(k,l+1)}\to 1.$$

This conjecture predicts that, except in the case $(k,l)=(1,1)$, almost all $(k+1,l)$-uniform matroids are $(k,l)$-uniform, or dually almost all $(k,l+1)$-uniform matroids are $(k,l)$-uniform. It is motivated by the conjecture that almost all matroids are paving; for $(k,l)=(2,1)$, the statement is weaker than that paving-matroid conjecture.