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