Pin's generalized synchronizing automaton conjecture

Let $A=(Q,\Sigma)$ be a finite automaton with $n=|Q|$ states. Its rank $r$ is the minimum rank of its input words, where the rank of an input word is the cardinality of the image of its associated mapping. A terminal word is an input word attaining this minimum rank. Pin's conjecture. Every $n$-state automaton of rank $r$ has a terminal word of length at most

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

This generalizes the synchronizing case, since rank-one automata are synchronizing and terminal words are then reset words. The supplied status evidence says that this generalized conjecture was disproved by Kari (2001).