Černý–Pin rank conjecture for complete automata

Let $\mathrsfs{A}$ be an $n$-state complete automaton, and let $r$ be the minimal rank of any word, where the rank of a word is the cardinality of its image on the state set. Rank conjecture. There exists a word of rank $r$ whose length is at most

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

This is a generalization of the Černý conjecture to nonsynchronizing automata; the case $r=1$ is the Černý conjecture. The stronger version originally stated by Pin was disproved, whereas this minimal-rank version is presented as an open conjecture.