Maximum-entropy extension conjecture for locally consistent marginals

Consider a set of locally consistent marginals satisfying the entropy conditions given by Eqs.

$$and$$

. Here, $S(X)$ denotes the entropy of the marginal on region $X$, and $A$, $B$, and $C$ are regions with unions denoted by $AB$, $BC$, and $ABC$. Maximum-entropy extension conjecture. The maximum-entropy state consistent with those marginals exists and obeys

$$S(ABC)=S(AB)+S(BC)-S(B)$$

for every $A$, $B$, and $C$ such that (i) $A$, $B$, $C$, $AB$, $BC$, and $ABC$ are all disk-like regions and (ii) $A$ and $C$ are not adjacent to each other. This conjecture would provide the reverse direction of the paper’s discussion: suitable local marginal data would determine a global state satisfying the expected entropy-scaling behavior. The source gives no resolution or proof, so the conjecture remains open.