  In this paper, we pursue our study of asymptotic properties of families of
random matrices that have a tensor structure. In previous work, the first- and
second-named authors provided conditions under which tensor products of unitary
random matrices are asymptotically free with respect to the normalized trace.
Here, we extend this result by proving that asymptotic freeness of tensor
products of Haar unitary matrices holds with respect to a significantly larger
class of states. Our result relies on invariance under the symmetric group, and
therefore on traffic probability.
  As a byproduct, we explore two additional generalisations: (i) we state
results of freeness in a context of general sequences of representations of the
unitary group -- the fundamental representation being a particular case that
corresponds to the classical asymptotic freeness result for Haar unitary
matrices, and (ii) we consider actions of the symmetric group and the free
group simultaneously and obtain a result of asymptotic freeness in this context
as well.
