Piecewise asymptotic formula for packing progressions

Let $M_k(n)$ denote the packing number with parameter $k$. Piecewise $M_k(n)$ conjecture. If $k\ll n$, then

$$M_k(n)\sim \frac{k^2}{2\ln k}\cdot n,$$

while otherwise,

$$M_k(n)\sim \frac{k^2}{2\ln k}\cdot n-\frac{k^3}{3\ln k}.$$

The source presents these as plausible asymptotics for $M_k(n)$ when $k<n$, motivated by a preceding theorem; they remain open.