Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected? That is for any pair of distinct states $p,q$ exists a positive integer $k$ such that either $p.v^k = q$ or $q.v^k = p$. Additionaly we can assume that there exists a word $w$ such that its underlying graph is connected.

The statement also can be reformulated in terms of linear algebra.