Černý's conjecture for D3-directing complete nondeterministic finite automata

Let a complete nondeterministic finite automaton (CNFA) be an automaton in which every state and input symbol has at least one outgoing transition. A word is D3-directing if, starting from every state $q$, there exists a path labelled by that word that ends in one common state $q_s$. Let $n$ be the number of states of the CNFA.

Černý's conjecture for D3-directing CNFAs. Every D3-directing CNFA with $n$ states admits a D3-directing word of length at most

$$(n-1)^2.$$

The paper states that this formulation is equivalent to Černý's conjecture for DFAs. The result would give the sharp quadratic bound for shortest D3-directing words; the conjecture remains unresolved.