The conjecture on non-synchronizing ranks of primitive groups

Let $G$ be a primitive permutation group of degree $n$, and define

$$\operatorname{NS}(G)=\{r:\text{there exists a map of rank }r\text{ not synchronized by }G\}$$

to be its set of non-synchronizing ranks. Non-synchronizing-rank conjecture.

$$|\operatorname{NS}(G)|=o(n).$$

This predicts that primitive groups have relatively few non-synchronizing ranks, contrasting with the linear lower bound established for transitive imprimitive groups. The supplied status is unknown, so the conjecture is recorded as open.