Let us consider the Caro's Acid: $\mathrm{H_2SO_5 \equiv SO_3H-OOH}$. It is widely known that the hydrogen tied with the $\mathrm{-OO-}$ group is pretty much unlikely to dissociate: $$ \text {Terminal dissociation: }\mathrm{SO_3H - OOH \rightleftharpoons H^+ + [SO_3-OOH]^-} \\ \text{An unlikely step: }\mathrm{[SO_3-OOH]^- \rightleftharpoons H^+ +[SO_3-OO]^{2-}} $$ The first time I encountered that fact I was presented a pretty inconclusive piece of argumentation: it was stated that $ \mathrm{-OO-} $ group is way more electronegative than $ \mathrm{-O-} $, which somehow brings about an obstacle for the dissociation.
Unfortunately, it was a long time ago and I cannot remember the exact details. I stumbled upon this problem not so long ago. Having a better background in chemistry, I attempted to apply the Sanders' electronegativity balance theory, but the result was pretty inane in terms of our chemical reality: $\chi_{eq}(\mathrm{H_2SO_4}) \approx 2.89894; \chi_{eq}(\mathrm{H_2SO_5}) \approx 2.96803 $. In other words: $ \chi_{eq}(\mathrm{H_2SO_5}) > \chi_{eq}(\mathrm{H_2SO_4}) $.
What am I doing wrong? Is it possible to tweak my attempt to use Sanders' theory? Is there any other approach to this problem?