The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$
It is known that the max-relative entropy is quasi-convex. That is, for $\rho=\sum_{i \in I} p_{i} \rho_{i}$ and $\sigma=\sum_{i \in I} p_{i} \sigma_{i}$ where $\rho_i, \sigma_i$ are quantum states and $p$ is a probability vector, it holds that
$$D_{\max }(\rho \| \sigma) \leq \max _{i \in I} D_{\max }\left(\rho_{i} \| \sigma_{i}\right).$$
I now look at a triparite state $\rho_{ABC} = \sum_i p_i\rho_{ABC}^i$. Is there any $i$ for which both the following inequalities hold?
$$D_{\max}(\rho_{ABC}\|\rho_{AB}\otimes\rho_C) \leq D_{\max}(\rho_{ABC}^i\|\rho_{AB}^i\otimes\rho_C^i)$$
$$D_{\max}(\rho_{AC}\|\rho_{A}\otimes\rho_C) \leq D_{\max}(\rho_{AC}^i\|\rho_{A}^i\otimes\rho_C^i)$$
Each individual equation holds due to quasi-convexity but jointly, it is not clear how to make them both hold for one $i$.
