The reconstructed kinematics of the isotropically emitted K0K0 and their decay vertex following a pp​ annihilation at rest allows to select events with specific neutral kaon topology {\it and} energy. Thus, from the accumulated events with CPLEAR one can define high statistics subgroups of any gravitationally oriented neutral kaons with ∼±60 cm maximum vertical displacement, in order to search for (anti)gravity effects. The suggested new data analysis allows to combine for the first time upward moving K0's with downward emitted overlineK0's, and {\it vice versa}, in order to search for any directional dependence of the CP parameters. The achieved limit for the relative K0−overlineK0 mass difference (∼10−18) may be indicative of a probably unique sensitivity for the proposed antigravity search. Existing anomalous energy dependent CP results along with the used non-horizontal kaon beam (having in fact an equal vertical kaon displacement per τS​ as in CPLEAR) are in favour of this proposal whatever the theoretical interpretation