Well-definedness of ADM mass, energy, linear and angular momentum

In asymptotically flat initial data sets, one can define the ADM mass, energy, linear and angular momenta via spatial limits of integrals over 2-spheres.
My questions are: is asymptotical flatness enough to ensure that the ADM mass, energy and linear momentum converge? (I know that for the angular momentum, that is not true). Does convergence of one of them imply convergence of the others?

And regarding the angular momentum, is being asymptotically Euclidean enough?