Within the Standard Model, the branching fraction of the rare decay \( {B}_s\to {K}^0{\overline{K}}^0 \) is related to other decay rates and CP asymmetries through the approximate SU(3) flavor symmetry of the strong interactions and the heavy-quark limit. Three such relations were shown to be violated at a level of about 3σ each. By means of a systematic search for new-physics explanations of these puzzles, we find that possible solutions are highly fine-tuned and either excluded by other data or rather implausible. The tight correlation between \( {B}_s\to K\overline{K} \) and B → πK decays, which is maintained even in the presence of flavor-specific new-physics operators, plays a central role in our analysis.