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The Standard-Model (SM) prediction for the CP-violating quantity$\epsilon_K^{\prime}/\epsilon_K$ deviates from its measured value by 2.8$\sigma$. It has been shown that this tension can be resolved within theMinimal Supersymmetric Standard Model (MSSM) through gluino-squark boxdiagrams, even if squarks and gluinos are much heavier than 1 TeV. The raredecays $K_L \to \pi^0\nu\bar{\nu}$ and $K^+ \to \pi^+\nu\bar{\nu}$ aresimilarly sensitive to very high mass scales and the first one also measures CPviolation. In this article, we analyze the correlations between$\epsilon^{\prime}_K/\epsilon_K$ and $B(K_L \to \pi^0\nu\bar{\nu})$ and $B(K^+\to \pi^+\nu\bar{\nu})$ within the MSSM aiming at an explanation of$\epsilon_K^{\prime}/\epsilon_K$ via gluino-squark box diagrams. The dominantMSSM contribution to the $K \to \pi\nu\bar{\nu}$ branching fractions stems frombox diagrams with squarks, sleptons, charginos, and neutralinos, and thepattern of the correlations is different from the widely studied $Z$-penguinscenarios. This is interesting in light of future precision measurements byKOTO and NA62 at J-PARC and CERN, respectively. We find $B(K_L \to\pi^0\nu\bar{\nu})/B^{SM} (K_L \to \pi^0\nu\bar{\nu})\lesssim 2\,(1.2)$ and$B(K^+ \to \pi^+\nu\bar{\nu})/B^{SM}(K^+ \to \pi^+\nu\bar{\nu}) \lesssim1.4\,(1.1)$, if all squark masses are above 1.5 TeV, gaugino masses obey GUTrelations, and if one allows for a fine-tuning at the $1\%\,(10\%)$ level forthe gluino mass. Larger values are possible for a tuned CP violating phase.Furthermore, the sign of the MSSM contribution to $\epsilon_K^{\prime}$ imposesa strict correlation between $B(K_L \to \pi^0\nu\bar{\nu})$ and the hierarchybetween the masses $m_{\bar{U}}$, $m_{\bar{D}}$ of the right-handed up-squarkand down-squark: sgn$[B(K_L \to \pi^0\nu\bar{\nu})-B^{SM} (K_L \to\pi^0\nu\bar{\nu})] = $sgn$(m_{\bar{U}}-m_{\bar{D}}) $.

$K\to \pi \nu\overline{\nu}$ in the MSSM in Light of the $\epsilon^{\prime}_K/\epsilon_K$ Anomaly