  Within the Standard Model, the branching fraction of the $B_s\to
K^0\overline{K}{}^0$ decay is related to other decay rates and CP asymmetries
through the approximate $SU(3)$ flavor symmetry of the strong interactions.
Three such relations were shown to be violated at a level of about $3\sigma$
each. By means of a systematic search for new physics explanations of these
puzzles, we find that possible solutions are highly fine-tuned and rather
implausible.
