Direct test of time-reversal symmetry in the entangled neutral kaon system at a $\phi$-factory

We present a novel method to perform a direct T (time reversal) symmetry test in the neutral kaon system, independent of any CP and/or CPT symmetry tests. This is based on the comparison of suitable transition probabilities, where the required interchange of in <-> out states for a given process is obtained exploiting the Einstein-Podolsky-Rosen correlations of neutral kaon pairs produced at a $\phi$-factory. In the time distribution between the two decays, we compare a reference transition like the one defined by the time ordered decays $(\ell^-,\pi\pi)$ with the T -conjugated one defined by $(3\pi^0, \ell^+)$. With the use of this and other T conjugated comparisons, the KLOE-2 experiment at DA$\Phi$NE could make a significant test.