Staircase characterization conjecture for extremal star-forest decompositions

Let $k$ be a positive integer, and let $G$ be a complete geometric graph on $2k$ points. A $k$-staircase is the point-set construction defined earlier in the source. Staircase characterization conjecture. If $G$ admits a decomposition into $k+1$ plane star-forests and the convex hull of $V(G)$ has size $k+1$, then $V(G)$ can be described as a $k$-staircase. The source reports that computations on point sets with $6$ and $8$ points found no counterexamples, but gives no proof or resolution. Thus the characterization remains open.