Hereditariness conjecture for the Černý bound

Let $\mathcal{A}$ be a synchronizing automaton, and let $\sigma\in\operatorname{Cong}(\mathcal{A})$ be a non-trivial congruence, so that the quotient automaton $\mathcal{A}/\sigma$ is defined. Hereditariness conjecture for the Černý bound. If the Černý conjecture holds for all simple automata—automata whose only congruences are the identity and universal relations—then it holds for all automata in general.

This is posed as a reduction of the Černý conjecture through quotients and congruences: proving the bound for simple automata would suffice to prove it for every automaton. The paper presents it as a main open problem.