Asymptotic sharpness of the trivial lower bound for fixed-length progression packing

Let $M_k(n)$ be the packing number for the progressions $B_d=\{d,2d,\ldots,nd\}$ with $d=1,2,\ldots,k$, with $n$ fixed. Fixed-length progression-packing conjecture. For every fixed $n$ and $k$ tending to infinity, we have

$$M_k(n)=(1+o(1))nk.$$

The source says this would make the trivial lower bound $nk$ asymptotically correct; the conjecture is open.