Corrected Guy–Kelly asymptotic conjecture for the no-three-in-line problem

Let $S_n$ be the set of $n^2$ points in $\mathbb{R}^2$ with integer coordinates $(x,y)$ satisfying $1\leq x,y<n$, and let $f_n$ be the maximum size of a subset of $S_n$ containing no three collinear points. Corrected Guy–Kelly conjecture. As $n$ tends to infinity,

$$f_n\sim \left(\frac{\pi}{\sqrt{3}}\right)n.$$

This is the corrected conjectured upper-bound asymptotic arising from Gabor Ellmann's correction to the Guy–Kelly heuristic; the source contrasts it with the original constant $\left(2\pi^2/3\right)^{1/3}$ and notes that a derivation of the corrected bound appeared in recent work of Prellberg.