Černý's conjecture for synchronizing DFAs

Let $M=(Q,\Sigma,\delta)$ be a deterministic finite automaton, and let $n=|Q|$ be its order. A word is synchronizing if it maps every state of $Q$ to the same state.

Černý's conjecture. If $M$ is synchronizable, then $M$ has a synchronizing word of length at most

$$(n-1)^2.$$

Černý constructed, for every $n$, an $n$-state DFA whose shortest synchronizing word has exactly this length, so the conjecture asserts that his examples are extremal. It remains a central open question in automata theory.