The relaxed locally identifying coloring conjecture for free-twin split graphs

Let $G$ be a free-twin split graph, meaning that $G$ is a split graph with no pair of free twins. Write $\chi_{rlid}(G)$ for its relaxed locally identifying chromatic number and $\omega(G)$ for its clique number. Free-twin split graph conjecture. If $G$ is a free-twin split graph, then

$$\chi_{rlid}(G)\leq\omega(G)+1.$$

The conjecture proposes an improvement of the previously established upper bound $\chi_{rlid}(G)\leq\omega(G)+2$ for split graphs; the source gives no resolution.