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A deterministic finite automaton in which every non-empty set of statesoccurs as the image of the whole state set under the action of a suitable inputword is called completely reachable. It was conjectured that in each completelyreachable automaton with $n$ states, every set of $k>0$ states is the image ofa word of length at most $n(n-k)$. We confirm the conjecture for completelyreachable automata with two input letters satisfying certain restrictions onthe action of the letters.

Don's conjecture for binary completely reachable automata: an approach and its limitations