The Big-Line-Big-Chromatic-Number Conjecture for planar visibility graphs

Let $P$ be a finite set of points in the plane, and let $\mathcal{V}(P)$ be its visibility graph. A $(k-1)$-colouring assigns one of $k-1$ colours to each point. Big-Line-Big-Chromatic-Number Conjecture. For all integers $k\geq2$ and $\ell\geq1$ there is an integer $n$ such that if $P$ has at least $n$ points and is assigned one of $k-1$ colours, then $P$ contains $\ell$ collinear points or some visible pair receives the same colour; equivalently, $\chi(\mathcal{V}(P))\geq k$. This weakening follows from the Big-Line-Big-Clique Conjecture and is known for $k\leq5$ or $\ell\leq3$; the cases beyond these ranges remain open.