Prove high-energy growth bound for gravitational amplitudes from first principles
In plain words
All positivity arguments assume that scattering amplitudes do not grow too fast at high energy. For gravity this growth limit is assumed, not proven.
Precise statement
Prove that the $2 \to 2$ amplitude of a consistent quantum gravity theory satisfies $M(s,t)/s^{2} \to 0\ \text{as}\ s \to \infty\ \text{at fixed}\ t < 0$ (so dispersion relations with two subtractions hold), using only unitarity, analyticity, crossing and polynomial or subexponential boundedness, in $D \ge 4$. Haring and Zhiboedov (2022) derived this bound with the extra assumption that scattering at large impact parameter is controlled by known semiclassical physics; justifying or removing that assumption, and handling $D = 4$ infrared divergences, is the open part.
What would settle it
A proof of the $s^{2}$ bound from axioms that do not assume semiclassical large-impact-parameter behavior, or a counterexample amplitude.
Status in the literature
The 2022 derivation rests on the semiclassical large-impact-parameter assumption.