Cup product in bounded cohomology of negatively curved manifolds

Let M be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential 2-form \xi \in \Omega ^2(M) defines a bounded cocycle c_\xi \in C_b^2(M) by integrating \xi over straightened 2-simplices. In particular Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] proved that, when M is a closed hyperbolic surface, \Omega ^2(M) injects this way in H_b^2(M) as an infinite dimensional subspace. We show that the cup product of any class of the form [c_\xi ], where \xi is an exact differential 2-form, and any other bounded cohomology class is trivial in H_b^{\bullet }(M).