The plane star-forest covering conjecture for complete geometric graphs

A complete geometric graph is the straight-line graph on a finite point set in the plane, with an edge between every pair of vertices; a plane star-forest is a star-forest whose edges do not cross. For a positive integer $n$, consider complete geometric graphs with $n$ vertices.

Plane star-forest covering conjecture. There is no complete geometric graph with $n$ vertices that can be decomposed into fewer than

$$\lceil 3n/4\rceil$$

plane star-forests.

The conjecture asserts that the four-cluster construction described in the paper, which uses $3n/4$ plane star-forests, is optimal when the vertices are not required to be in convex position. The convex-position case is proved with the stronger lower bound $n-1$, while the general case remains open.