Convex-hull-size conjecture for optimal star-forest decompositions

Let $n\ge3$ be odd, and let $G$ be a complete geometric graph on $n$ vertices that can be decomposed into $\left\lceil\frac{n}{2}\right\rceil+1$ plane star-forests. Convex-hull-size conjecture. The convex hull of the vertex set $V(G)$ has size at most

$$\left\lceil\frac{n}{2}\right\rceil+1.$$

For even order, the source proves the analogous bound when $n=2k$, while the odd-order case is proposed as a conjecture. Its general validity remains open.