Clique-bounded degree conjecture for derangement graphs

Let $G$) be a transitive permutation group of degree $n$, and let $\Gamma_G$ denote its derangement graph. For a positive integer $c$, say that $\Gamma_G$ contains no clique of size $c$ when it has no complete subgraph on $c$ vertices. Clique-bounded degree conjecture. There exists a function $F$ such that, if $G$ is a transitive permutation group of degree $n$ whose derangement graph contains no clique of size $c$, then

$$n\leq F(c).$$

The conjecture formalizes the observation that excluding large cliques in derangement graphs should strongly restrict the degree of a transitive action; the supplied text gives no resolution status.