  The $K \rightarrow \mu\mu$ decays have recently been identified as another
golden kaon physics mode alongside the rare $K \rightarrow\pi\nu\bar{\nu}$
processes. These golden modes provide precision tests of the Standard Model
with very high sensitivity to New Physics. The presented study is exploring the
possibility to address the $K_L - K_S \rightarrow \mu^+\mu^-$ interference
experimentally and outlines the challenges associated with such an ambitious
project for the far future. A next-generation experiment at the intensity
frontier is required that should be capable of collecting a large sample of
$\mathcal{O}(10^{14} - 10^{15})$ $K_L$ and $K_S$ decays. Challenges related to
the beamline design and detector technology need to be overcome if we want to
address this mode experimentally. A significant background suppression of $K_S
\rightarrow \pi^+\pi^-$ and radiative $K_L \rightarrow \mu^+\mu^-\gamma$ decays
is imperative for a few $\%$ measurement, which would require excellent
kinematic resolution and efficient photon detection. The first attempt at a
possible experimental setup to measure this effect is presented. Last but not
least, a huge number of neutral particles produced offers the possibility to
study a plethora of other rare $K_L$, $K_S$ decays as well as hyperon decays
enhancing the physics motivation for such an initiative.
