I’m sure I’m missing your point, but FWIW my original claim was only about the (im)possibility of coordination on a non-Nash equilibrium solution (i.e. of coordinating on a solution that is not incentive-compatible). Coordinating on one of a number of Nash equilibria (which is the issue in battle of the sexes) is a different matter entirely (and not one I am claiming anything about).
I’m sure I’m missing your point, but FWIW my original claim was only about the (im)possibility of coordination on a non-Nash equilibrium solution (i.e. of coordinating on a solution that is not incentive-compatible). Coordinating on one of a number of Nash equilibria (which is the issue in battle of the sexes) is a different matter entirely (and not one I am claiming anything about).