Černý's conjecture for finite groups

Let $G$ be a finite group and let $cDelta$ be a generating set. The automaton $(G,cDelta)$ is the Cayley graph of $G$ with respect to $cDelta$; call $G$ a Černý group if every synchronizing automaton containing such a Cayley graph satisfies the Černý bound. Černý-group conjecture. All finite groups are Černý groups. This is posed as an open question motivated by the absence of a quadratic bound for arbitrary synchronizing automata; its resolution is not supplied in the paper.