We point out that using current knowledge of \( \mathcal{B}\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right) \) and \( \mathcal{B}\left({K}_L^0\to \gamma \gamma \right) \) , one can extract short-distance information from the combined measurement of the time-integrated CP asymmetry, ACP(K0 → μ+μ−), and of \( \mathcal{B}\left({K}_S^0\to {\mu}^{+}{\mu}^{-}\right) \) . We discuss the interplay between this set of observables, and demonstrate that determining sign[ACP(K0 → μ+μ−)] would eliminate the discrete ambiguity in the Standard Model prediction for \( \mathcal{B}\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right) \) . We then move on to feasibility studies within an LHCb-like setup, using both time-integrated and time-dependent information, employing K0 and \( {\overline{K}}^0 \) tagging methods. We find that, within an optimistic scenario, the short-distance amplitude, proportional to the CKM parameter combination \( \mid {A}^2{\lambda}^5\overline{\eta}\mid \) , could be constrained by LHCb at the level of about 35% of its Standard Model value, and the discrete ambiguity in \( \mathcal{B}{\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right)}_{\textrm{SM}} \) could be resolved at more than 3σ by the end of the high luminosity LHC.
