  Cosmological correlators from inflation are often generated at tree level and
hence loop contributions are bounded to be small corrections by perturbativity.
Here we discuss a scenario where this is not the case. Recently, it has been
shown that for any number of scalar fields of any mass, the parity-odd
trispectrum of a massless scalar must vanish in the limit of exact scale
invariance due to unitarity and the choice of initial state. By carefully
handling UV-divergences, we show that the one-loop contribution is
non-vanishing and hence leading. Surprisingly, the one-loop parity-odd
trispectrum is simply a rational function of kinematics, which we compute
explicitly in a series of models, including single-clock inflation. Although
the loop contribution is the leading term in the parity-odd sector, its
signal-to-noise ratio is typically bounded from above by that of a
corresponding tree-level parity-even trispectrum, unless instrumental noise and
systematics for the two observables differ. Furthermore, we identify a series
of loop contributions to the wavefunction that cancel exactly when computing
correlators, suggesting a more general phenomenon.
