Leveraging the nine-octahedra edge example for bounds on convex-curve universal point sets

Ascertain whether the existence of a planar graph formed by gluing eight regular octahedra to the faces of a central octahedron, which cannot be drawn so that all edges touch a convex curve, yields any implications or provable bounds on the minimum number of convex curves required to support universal point sets for planar graphs.

Background

The paper contrasts the situation for faces and edges: although all faces of any planar graph can be crossed by a smooth convex curve, the authors provide an example (a compound of nine octahedra) showing that not all planar graphs can have drawings where all edges are touched by a convex curve.

However, the authors explicitly state that they do not know how to use this edge-based obstruction to deduce anything about the number of convex curves needed to support universal point sets, leaving open whether such examples can imply lower bounds or other constraints on convex-curve support for universal point sets.