The relative entropy variance between two quantum states $\rho$ and $\sigma$ is defined to be
$$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$
where $D(\rho\|\sigma)$ is the quantum relative entropy.
Analogous to the mutual information, one can define the mutual information variance for a bipartite state $\rho_{AB}$ by choosing the second argument of the relative entropy variance to be the product of the marginals $\rho_A\otimes\rho_B$. The mutual information is bounded by $\log d^2$, where $d$ is the dimension of the registers $A$ and $B$. My question is if the mutual information variance is also bounded? That is,
$$V(\rho_{AB}\|\rho_A\otimes\rho_B) \leq \ ?$$
