Černý's conjecture for cyclic groups

Let $G$ be a finite cyclic 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 cyclic groups. All cyclic groups are Černý groups. This is motivated by Dubuc's work and is presented as an open conjecture; the paper notes that extending the argument may require number theory.