Continuity bounds on $D_{max}(rho_{AB}|rho_Aotimesrho_B)$

The max-relative entropy between two states is defined as

$$D_{max }(rho | sigma):=log min {lambda: rho leq lambda sigma},$$

where $rholeq sigma$ should be read as $sigma – rho$ is positive semidefinite. Consider the following quantity for a bipartite state $rho_{AB}$ with reduced states $rho_{A}$ and $rho_{B}$.

$$I_{max}(rho_{AB}) = D_{max}(rho_{AB}||rho_{A}otimesrho_{B})$$

I would like to know if this satisfies a continuity bound. That is, given $rho_{AB}approx_{epsilon}sigma_{AB}$ in some distance measure, can we bound $|I_{max}(rho_{AB}) – I_{max}(sigma_{AB})|$?

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Motivation for question: Recall the quantum relative entropy $D(rho||sigma) = text{Tr}(rhologrho – rhologsigma)$ with the convention that $0log 0 = 0$. Let us define the mutual information as follows

$$I(rho_{AB}) = D(rho_{AB}||rho_{A}otimesrho_{B}) = -S(rho_{AB}) + S(rho_A) + S(rho_B),$$

where $S(rho) = -text{Tr}(rhologrho)$ is the von Neumann entropy. In this case, we may use Fannes inequality to find a bound on $|I(rho_{AB}) – I(sigma_{AB})|$ in terms of $|rho_{AB} – sigma_{AB}|_1$. I’m wondering if the move from $D(.||.)$ to $D_{max}(.||.)$ can be made while still having some Fannes type bound.