  We consider the possible signatures of a recently introduced class of healthy
theories beyond Horndeski models on higher-order correlators of the
inflationary curvature fluctuation. Despite the apparent large number and
complexity of the cubic interactions, we show that the leading-order bispectrum
generated by the Generalized Horndeski (also called $G^3$) interactions can be
reduced to a linear combination of two well known $k$-inflationary shapes. We
conjecture that said behavior is not an accident of the cubic order but a
consequence dictated by the requirements on the absence of Ostrogradski
instability, the general covariance and the linear dispersion relation in these
theories.
