This is a conjecture of João Araújo, and is related to synchronization and to the Černý conjecture in automata theory.
Let G be a primitive permutation group on the finite set X, and let f be a map from X to itself which is not a permutation. G is said to be synchronizing if, for any such map f, the semigroup 〈G,f〉 contains a constant function.
It is known that a synchronizing group is primitive (preserves no non-trivial equivalence relation on X), but the converse is false.
The group G is said to be almost synchronizing if, for any map f whose kernel (the partition of X given by inverse images of points in the image) is non-uniform (has parts of different sizes), 〈G,f〉 contains a constant function. Again, it is true that an almost synchronizing group is primitive.
Prove or disprove the converse, that is, any primitive group is almost synchronizing.