Quadratic synchronizing-word bound conjecture

Let ${\cal M}\subset\{0,1\}^{n\times n}$ be a synchronizing automaton, with synchronizing words defined by products of matrices from ${\cal M}$ that have the form $e e_i^T$. Quadratic synchronizing-word bound conjecture. There is a fixed constant $K>0$ such that every synchronizing automaton has a synchronizing word of length at most $Kn^2$. This is a weaker version of Černý's conjecture, whose general case is open; the paper notes that even the existence of a quadratic bound with an unspecified constant was not known there.