On the invariance of the string topology coproduct

We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class built from homotopy data associated to a homotopy equivalence as well as the ``fake diagonal''. The vanishing of our obstruction class can be seen as a way to measure a form of boundedness for homotopy equivalences. We show that the same obstruction rules the failure of invariance for a generalization of the coproduct to higher dimensional loops.