  Averaged Null Energy Conditions (ANECs) hold in unitary quantum field
theories. In conformal field theories, ANECs in states created by the
application of the stress tensor to the vacuum lead to three constraints on the
stress-tensor three-point couplings, depending on the choice of polarization.
The same constraints follow from considering two-point functions of the stress
tensor in a thermal state and focusing on the contribution of the stress tensor
in the operator product expansion (OPE). One can observe this in holographic
Gauss-Bonnet gravity, where ANEC saturation coincides with the appearance of
superluminal signal propagation in thermal states. We show that, when this
happens, the corresponding generalizations of ANECs for higher-spin
multi-stress tensor operators with minimal twist are saturated as well and all
contributions from such operators to the thermal two-point functions vanish in
the lightcone limit. This leads to a special near-lightcone behavior of the
thermal stress-tensor correlators -- they take the vacuum form, independent of
temperature.
