Big line or big clique conjecture for planar point sets

Let $P$ be a finite set of points in the plane. Two distinct points $v,w\in P$ are visible with respect to $P$ if

$$P\cap\overline{vw}=\{v,w\},$$

where $\overline{vw}$ is the closed line segment between $v$ and $w$. The visibility graph of $P$ has vertex set $P$, with two distinct points adjacent exactly when they are visible.

Big line or big clique conjecture. For all integers $k$ and $\ell$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains either $\ell$ collinear points or $k$ pairwise visible points; equivalently, its visibility graph contains a $k$-clique.

This conjecture predicts a dichotomy between arbitrarily large collinear subsets and arbitrarily large cliques in the visibility graph of a planar point set. The source attributes it to KPW; its resolution is not established by the supplied context.