The generic MPDO conjecture on exponential decay of conditional mutual information

A Matrix Product Density Operator (MPDO) is a one-dimensional tensor-network representation of a mixed quantum state. For a tripartite state $\rho_{ABC}$, the conditional mutual information is

$$I(A:C|B)_\rho:=S(AB)_\rho+S(BC)_\rho-S(B)_\rho-S(ABC)_\rho,$$

where $S(A)_\rho=-\operatorname{Tr}\rho_A\log_2\rho_A$.

Generic MPDO conjecture. For a `generic' MPDO, there exists a constant $c>0$ such that for any tripartition $ABC$ in which $B$ separates $A$ from $C$ by distance $\ell$,

$$I(A:C|B)=O(e^{-c\ell}).$$

Exponential decay of conditional mutual information is motivated by the characterization of states that can be well approximated by local Gibbs states. The source deliberately leaves the meaning of `generic' ambiguous, and no resolution is given.