Approximate quantum Markov property for Gibbs states

Let $\rho_\beta$ be an arbitrary quantum Gibbs state on a system partitioned as $\Lambda=A\sqcup B\sqcup C$, and let $d_{A,C}$ denote the distance between $A$ and $C$. Write ${\mathcal I}_{\rho_\beta}(A:C\mid B)$ for the conditional mutual information. Approximate quantum Markov conjecture. The conditional mutual information rapidly decays with the separation $R=d_{A,C}$:

$${\mathcal I}_{\rho_\beta}(A:C\mid B)\leq {\mathcal G}_{\mathcal I}(R).$$

Here ${\mathcal G}_{\mathcal I}(R)$ is super-polynomially decaying and may depend on $\beta$, the regions $\{A,B,C\}$, and the fundamental parameters listed in the paper. This conjecture seeks a quantum analogue of the Hammersley--Clifford relation between Markov structure and Gibbs states. It is known for short-range commuting Hamiltonians at arbitrary temperatures, whereas the corresponding exact Markov property fails for non-commuting Hamiltonians; the approximate statement for general quantum Gibbs states remains open.