The three-color homothetic-copy conjecture for convex sets

Let $D$ be a convex set in the plane, and call a scaled and translated copy of $D$ without rotation a homothetic copy of $D$. Let $S$ be a finite set of points. The three-color homothetic-copy conjecture. For every plane convex set $D$ there is an $m$ such that any finite set of points admits a $3$-coloring such that any homothetic copy of $D$ containing at least $m$ points contains two points with different colors. The disk case is stated to remain open, while the source proves the conjecture for convex polygons.