 Solution

Yes. Write $D_S=D(Q_{X_S}\Vert P_{X_S})$ and $D=D_{\{1,\dots ,n\}}$. The bounds from b and d are respectivelyThe Strong form of Han's entropy inequality, obtained by repeated entropy submodularity, statesAfter replacing entropies by divergences from the product reference, whose cross-entropy terms cancel, this is exactly $B_{\mathrm{new}}\le B_{\mathrm{old}}$. Equality holds for product $Q$ and when $n=2$; dependence can make the new bound strictly smaller.