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
$$D_{\max }(\rho \| \sigma) \leq \max _{i \in I} D_{\max }\left(\rho_{i} \| \sigma_{i}\right).$$
Is there a lower bound for $D_{\max }(\rho \| \sigma)$ in terms of $D_{\max }(\rho_i \| \sigma_i)$?