Let $X_1$ $X_2$ and $X_3$ finite sets and let $\pi_{1,2} \in \Delta(X_1 \times X_2)$, $ \pi_{1,3} \in \Delta(X_1 \times X_3)$ and $\pi_{2,3} \in \Delta(X_2 \times X_3)$ probability measures that satisfy the following:
For each different indexes $i,j,k \in \{1,2,3\}$ and each $Y_i \subset X_i$, $Y_j \subset X_j$ and $Y_k \subset X_k:$ $$ \pi_{ij}(Y_i \times Y_j) \leq \pi_{ik}( Y_i \times Y_k) + \pi_{jk}(Y_j \times Y_k^c) $$
Is it always possible to construct a measure $\pi \in \Delta(X_1\times X_2 \times X_3)$ such that for each different indexes $i,j \in \{1,2,3\}$, $ \pi_{ij} = Marg_{X_i \times X_j} \pi$ ?