Clique-index conjecture for decompositions of complete geometric graphs

Let $\mathsf{K}_n$ be a complete geometric graph of order $n$, and let $[\mathsf{K}_n,P]$ be a decomposition. The clique index $\omega'([\mathsf{K}_n,P])$ is the maximum size of a pairwise intersecting family of elements of $P$. Clique-index conjecture. Every decomposition satisfies

$$\omega'([\mathsf{K}_n,P])\leq \frac{n^2}{9}+\Theta(n).$$

This is presented as a weaker conjecture because $\omega'([\mathsf{K}_n,P])\leq\chi'([\mathsf{K}_n,P])$; the proposed bound remains open.