Asymptotic formula for packing progressions of intermediate length

Let $m_k(n)$ denote the packing number for progressions with parameter $k$. Intermediate-length packing conjecture. For every $k$ satisfying $\Omega(\sqrt{n})=k<\sqrt{n}$, we have

$$m_k(n)\sim \left(1-\frac{k^2}{3n}\right)\frac{k}{\ln k}\cdot n.$$

The source identifies this as the remaining case after the conjecture for $m(n)$ would settle the range $k\geq\sqrt n$; it is left open.