The asymptotic enumeration conjecture for dope matrices

Let $\mathcal{D}_n^m$ denote the set of dope matrices with parameters $n$ and $m$. For functions $m(t),n(t)\colon \mathbb Z_{>0}\to\mathbb Z_{>0}$ satisfying

$$n(t)\to\infty,\qquad 1<m(t)\leq \frac{n(t)^2+n(t)}{2},$$

Asymptotic enumeration conjecture. We have

$$\log\left\lvert\mathcal{D}_n^m\right\rvert\sim \log\left(n^m\binom{mn}{n}\right).$$

This conjecture asserts that the lower bound in the paper's asymptotic estimate gives the correct asymptotic formula throughout the stated range of parameters. Its resolution is not indicated in the supplied text.