Big line, big clique conjecture for planar visibility graphs

Let $X$ be a finite set of points in the Euclidean plane. For $u,v\in X$, let $(uv)$ denote the open line segment with endpoints $u$ and $v$, and let the visibility graph $G_X$ be the graph with vertex set $X$ in which $u$ and $v$ are adjacent exactly when $(uv)$ contains no point of $X$. Big line, big clique conjecture. For any fixed $\ell$ and $k$ there is a constant $c=c(k,\ell)$ such that every finite planar point set of size at least $c$ has either $\ell$ collinear points or a visibility graph with a clique of size $k$. This conjecture is currently open for all $k\geq 6$ and $\ell\geq 4$; finiteness is necessary because infinite point sets provide counterexamples.