Length of a synchronizing word of finite automata

Question

What is the maximal length of a shortest synchronizing word over an automaton.

Lower bound is asymptotically $n^2$.

Lower bound construction

Upper bound is asymptotically $n^3$. Consider the shortest synchronizing word W for an automata.

Upper bound proof

A Note on Homogeneous Experiments with Finite Automata, 1964 ↗
Pin 1983 ${n^3-n \over 6}$ upper bound
$n^2$ was shown for few classes of synchronizing automata
On two combinatorial problems arising from automatatheory, 1981 ↗
Kari 2003, if the underlying digraph is Eulerian, then the minimum reset word is of length at most (n-2)(n-1)+1 (open of tight?)
Trahtman 2011 ${7n^3+12n^2-4n \over 48}$