The conjecture that primitive groups synchronize every non-uniform map

Let $T_n$ be the full transformation semigroup on an $n$-element set. For a transformation $a\in T_n$, call it non-uniform if its kernel classes are not all of the same size. A permutation group $G$ is primitive if it preserves no nontrivial partition of the underlying set, and it synchronizes $a$ when the semigroup generated by $G$ and $a$ contains a map of rank $1$.

Synchronization conjecture. Every primitive permutation group synchronizes every non-uniform map.

This is presented as the biggest open problem in the area. It concerns the interaction between primitivity and synchronization in transformation semigroups, and its general validity remains unresolved.