On a mixed-state extension of the holographic signal inequality

A novel inequality involving the residual entropy and genuine multi-entropy was proposed in \cite{Balasubramanian:2025hxg} for tripartite holographic pure states, using which it was argued, that purely GHZ-like tripartite entanglement is not allowed in holography. In this work, we generalize this holographic signal inequality to mixed states. In a minimal extension, we compute the reflected genuine multi-entropy following \cite{Yuan:2024yfg} and find a class of holographic geometries that violate this minimally extended inequality due to vanishing Markov gap. We can symmetrize this prescription, where instead of computing the residual entropy on the given mixed state $\rho_{ABC}$, we compute it on its canonical purification. The inequality is restored on the canonically purified state, as expected. Finally, we conjecture a new inequality for tripartite holographic states and give supporting evidence.