The convex-polygon realization conjecture for maximum rectilinear crossing numbers

Let $G$ be any graph. A rectilinear drawing represents the vertices of $G$ by points in the plane and its edges by straight-line segments. A convex-polygon drawing is one in which all vertices are vertex points of a convex polygon.

Convex-polygon realization conjecture. The maximum rectilinear crossing number of $G$ can be realized in a convex-polygon drawing.

If true, this would reduce the search for maximum rectilinear crossing numbers to drawings with all vertices in convex position. The paper presents this as a general conjecture useful for determining the maximum rectilinear crossing number.