Connection between smooth max-relative entropy and smooth max-information

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. There is also a smoothed version of this quantity and this is given by taking the infimum of $D_{max}(rho|sigma)$ over all states $bar{rho}$ which are within an $varepsilon$ ball of $rho$ according to some metric. For example, one may require that the trace distance between $rho$ and $bar{rho}$ is at most $varepsilon$ and define a ball this way. So we have

$$D^{varepsilon}_{max}(rho|sigma) = inflimits_{bar{rho}inmathcal{B}^{varepsilon}(rho)}D_{max}(bar{rho}|sigma)$$

Now consider the case where $rho = rho_{AB}$ (some bipartite state) and $sigma = rho_Aotimesrho_B$. A quantity known as the max-information that $B$ has about $A$ is given by

$$I_{max}(A:B)_rho = D_{max}(rho_{AB}||rho_Aotimesrho_B)$$

Note that this is not the only definition of the max-information (there are several as shown here but they are all equivalent after smoothing). The smoothed max-information for our definition of the max-information is

$$I^{varepsilon}_{max}(A:B)_{rho_{AB}} = inflimits_{bar{rho}_{AB}inmathcal{B}^{varepsilon}(rho_{AB})} I(A:B)_{bar{rho}} = inflimits_{bar{rho}_{AB}inmathcal{B}^{varepsilon}(rho_{AB})}D_{max}(bar{rho}_{AB}|bar{rho}_Aotimesbar{rho}_B)$$

In contrast, the smoothed max-relative entropy is

$$D^{varepsilon}_{max}(rho_{AB}|rho_Aotimesrho_B) = inflimits_{bar{rho}_{AB}inmathcal{B}^{varepsilon}(rho_{AB})} D_{max}(bar{rho}_{AB}|rho_Aotimesrho_B)$$

Are the two quantities $D^{varepsilon}_{max}(rho_{AB}|rho_Aotimesrho_B)$ and $I^{varepsilon}_{max}(A:B)_{rho_{AB}}$ close to each other (e.g. the difference is some function of $varepsilon$) such that they are qualitatively equivalent?