Constant number of convex curves for vertex placements of all planar graphs

Determine whether there exists a constant k such that for every planar graph G there is a straight-line planar drawing in which all vertices of G lie on the union of at most k convex curves in the plane.

Background

A central motivation of the paper is the paper of universal point sets for planar graphs and whether they can be supported by a bounded number of convex curves, analogous to prior work on supporting universal sets with lines. While the paper proves that every planar graph has a straight-line drawing whose faces are all crossed by a smooth convex curve, it does not settle whether a constant number of convex curves suffices to place all vertices of every planar graph.

Existing results with lines show that an unbounded number of lines is required to support all planar graphs, but it remains unknown if replacing lines by convex curves reduces this complexity to a constant. This question is stated explicitly as the most salient remaining open problem.