  Chern-Simons gravity coupled to the scalar sector through a generic coupling
function $f(\phi)$ can be tested at the very high energies of the inflationary
period. In 1706.04627, we computed the theoretical parity breaking signatures
of the $\langle \gamma \gamma \zeta \rangle$ primordial bispectrum which mixes
two gravitons and one scalar curvature perturbation. We defined a parameter
$\Pi$ which measures the level of parity breaking of the corresponding
bispectrum. In this work we forecast the expected $1 \sigma$ error on $\Pi$
using the cosmic microwave background (CMB) angular bispectra. We find that,
given the angular resolution of an experiment like $Planck$, $\Pi \sim 10^6$ is
detectable via the measurement of $BBT$ or $BBE$ angular bispectra if the
tensor-to-scalar ratio $r = 0.01$. We also show that, from the theoretical
point of view, $\Pi$ can be greater than $10^6$. Thus, our conclusion is that
$BBT$ or $BBE$ CMB angular bispectra can become an essential observable for
testing Chern-Simons gravity in the primordial universe.
