  It has been shown that primordial tensor non-Gaussianities from a cubic Weyl
action with a non-dynamical coupling are suppressed by the so-called slow-roll
parameter in a conventional framework of slow-roll inflation. In this paper, we
consider matter bounce cosmology in which the background spacetime is no longer
quasi-de Sitter, and hence one might expect that the matter bounce models could
predict non-suppressed non-Gaussianities. Nevertheless, we first show that the
corresponding non-Gaussian amplitudes from the cubic Weyl term with a
non-dynamical coupling are much smaller than those from the conventional
slow-roll inflation, in spite of the fact that there is no slow-roll
suppression. We then introduce a dynamical coupling that can boost the
magnitude of graviton cubic interactions and clarify that there is a parameter
region where the tensor non-Gaussianities can be enhanced and can potentially
be tested by cosmic microwave background experiments.
