  Recently, Brassard et. al. conjectured that the fact that the maximal
possible correlations between two non-local parties are the quantum-mechanical
ones is linked to a reasonable restriction on communication complexity. We
provide further support for the conjecture in the multipartite case. We show
that any multipartite communication complexity problem could be reduced to
triviality, had Nature been more non-local than quantum-mechanics by a quite
small gap for any number of parties. Intriguingly, the multipartite
nonlocal-box that we use to show the result corresponds to the generalized Bell
inequality that manifests maximal violation in respect to a local
hidden-variable theory.
