The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is positive semidefinite. In other words, $D_{\max}$ is the logarithm of the smallest positive real number that satisfies $\rho\leq\lambda\sigma$. I would like to understand the following properties of this quantity when the states are bipartite i.e. they live on $H_A\otimes H_B$. In the following, all $\rho$ and $\sigma$ correspond to quantum states (positive semidefinite matrices with unit trace).
A quantity known as the max-information that $B$ has about $A$ is given by
$$I_{\max}(A:B)_\rho = \min\limits_{\sigma_B} D_{\max}(\rho_{AB}||\rho_A\otimes\sigma_B)$$
Let the minimum on the right hand side be achieved by the state $\sigma^\star_B$. My questions are as follows
Can someone provide an example of a state $\rho_{AB}$ for which $\sigma^\star_B \neq \rho_B$?
Is it true that $D_{\max}(\rho_{B}||\sigma^\star_B) \leq D_{\max}(\rho_{B}||\sigma_B)$ for all $\sigma_B$ i.e. is the $D_{\max}$ minimizing state preserved under a partial trace?