Černý's conjecture for abelian groups

Let $G$ be a finite abelian group. Call $G$ a Černý group if the relevant synchronizing automata containing its Cayley graph admit synchronizing words satisfying the Černý bound. Černý-group conjecture for abelian groups. All abelian groups are Černý groups. This is proposed as the next natural step after the cyclic case and remains open in the supplied text.