Before revision:  In 2018, Caraballo, D\'iaz-B\'a{\~n}ez, Fabila-Monroy, Hidalgo-Toscano, Lea{\~n}os, Montejano showed that n ^{ 3n-o (n)} order types can be realized on an integer grid of polynomial size.
After revision:  In 2018, Caraballo, D\'iaz-B\'a{\~n}ez, Fabila-Monroy, Hidalgo-Toscano, Lea{\~n}os, Montejano showed that at least n ^{ 3n-o (n)} order types can be realized on an integer grid of polynomial size.