Rank-two word conjecture for irreducible automata

Let $\mathcal{A}=(Q,\Sigma,\delta)$ be an irreducible automaton. For a word $u\in\Sigma^*$, let $\operatorname{rk}(u)$ denote the cardinality of the image of $Q$ under $u$. Rank-two word conjecture. There exists a word $u\in\Sigma^*$ such that

$$\operatorname{rk}(u)=2.$$

All irreducible examples constructed in the paper have former-rank two, which motivates the conjecture. The source presents it as unresolved and notes that simple, non-irreducible automata of former-rank two also exist, so rank two alone cannot characterize irreducibility.