The monochromatic-line conjecture for coloured planar point sets

Let $P$ be a finite set of points in the plane, and colour each point with one of $k$ colours. A monochromatic line means a maximal set of collinear points in $P$ whose points all receive the same colour. Monochromatic-line conjecture. For all positive integers $k$ and $\ell$ there is an integer $n$ such that every set $P$ of at least $n$ points in the plane contains $\ell$ collinear points or a monochromatic line. This is presented as a strengthening of the Big-Line-Big-Chromatic-Number Conjecture; no resolution is given in the source.