  We show that the contribution of the primordial trispectrum to the energy
density of the scalar-induced stochastic gravitational wave background cannot
exceed the one from the scalar power spectrum in conventional inflationary
scenarios. Specifically, we prove in the context of scale-invariant theories
that neither regular trispectrum shapes peaking in so-called equilateral
configurations, nor local trispectrum shapes diverging in soft momentum limits,
can contribute significantly. Indeed, those contributions are always bound to
be smaller than an order-one (or smaller) number multiplying the relative
one-loop correction to the scalar power spectrum, necessarily much smaller than
unity in order for the theory to be under perturbative control. Since a no-go
theorem is only worth its assumptions, we also briefly discuss a toy model for
a scale-dependent scalar spectrum, which confirms the robustness of our no-go
result.
