Asymptotic formula for packing arithmetic progressions

Let $m(n)$ denote the maximum size of a packing of arithmetic progressions in the setting of the paper. Asymptotic packing conjecture.

$$m(n)\sim \frac{4}{3}\cdot \frac{n^{3/2}}{\ln n}.$$

This conjecture seeks the leading constant in the asymptotic growth of $m(n)$; the source states that additional ideas are needed and leaves the problem open.