  Tensor non-Gaussianities are a key ingredient to test the symmetries and the
presence of higher spin fields during the inflationary epoch. Indeed, the shape
of the three point correlator of the graviton is totally fixed by the
symmetries of the de Sitter stage and, in the case of parity conservation, gets
contributions only from the ordinary gravity action plus a higher derivative
term called the (Weyl)$^3$ action. We discuss current and future bounds on the
three point tensor contribution from the (Weyl)$^3$ term using cosmic microwave
background (CMB) bispectra. Our results indicate that forthcoming experiments,
such as LiteBIRD, CMB-S4 and CORE, will detect the presence of the (Weyl)$^3$
term if $M_p^4 L^4 \sim 10^{17} r^{-4}$, where $L$ parametrizes the strength of
the (Weyl)$^3$ term and $r$ is the tensor-to-scalar ratio, which corresponds to
$L\gtrsim 3.2 \times 10^5 M_p^{-1}$, while the current upper limit is $M_p^4
L^4 = (1.1 \pm 4.0) \times 10^{19} r^{-4}$ (68%CL).
