Černý's conjecture for dihedral groups

Let $G$ be a finite dihedral 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 dihedral groups. Dihedral groups are Černý groups. The source motivates this by suggesting an extension of Dubuc's techniques to Cayley graphs generated by a reflection and a rotation, and presents the claim as a bolder open conjecture.