A trace inequality for Euclidean gravitational path integrals

It has recently been shown that any Euclidean gravitational path integral satisfying a simple set of axioms defines type I von Neumann algebras of bulk observables acting on compact closed codimension-2 asymptotic boundaries. These axioms imply the validity of an inequality that, when the gravitational theory admits an holographic dual CFT, on the CFT side takes the form $Tr(AB) \le Tr(A)Tr(B)$ for any two positive operators $A$ and $B$. As a consistency check, we prove that the Euclidean gravitational path integral respects such inequality at all orders in the semi-classical expansion and with arbitrary higher-derivative corrections. The argument relies on a conjectured property of the classical gravitational action, which in particular implies a positive action conjecture for quantum gravity wavefunctions that we prove for Jackiw-Teitelboim gravity and motivate for more general theories.