Asymptotic formula for the maximum packing number

Let $M(n)$ denote the maximum packing number in the setting of the paper. Maximum-packing conjecture.

$$M(n)\sim \frac{1}{6}\cdot \frac{n^{3}}{\ln n}.$$

The conjecture predicts the leading constant for $M(n)$; the source presents it as an open asymptotic problem and relates the expected factor to a preceding theorem.