The large-rank almost synchronizing conjecture for primitive groups

Let $G$ be a primitive permutation group of degree $n$, and let a map have rank $r$, meaning that its image has cardinality $r$. Large-rank almost synchronizing conjecture. If

$$\frac{n}{2}<r<n,$$

then $G$ synchronizes the map. Equivalently, every map of rank strictly between $n/2$ and $n$ is synchronized by $G$; such maps are non-uniform. The paper presents this as a weaker version of the almost synchronizing conjecture and states that it remains a difficult problem to resolve the broader classification question.