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The elastic production of top quark pairs in $pp$ collisions at low and high luminosity regimes is investigated in detail. We extend the study performed in Phys. Rev. D102, 074014 (2020) which has demonstrated that the sum of two semi-exclusive $t\bar{t}$ production modes, namely in photon-Pomeron and Pomeron-Pomeron interactions, can be experimentally measured when the $t\bar{t}$ system decays semi-leptonically, $t\bar{t}\rightarrow jjblν_l\bar{b}$, both forward protons are tagged and a low amount of pile-up is present. In this study we focus on separating individual channels and a special attention is paid to the situation at high-luminosity LHC environment. We observe that the separation of the pomeron-pomeron and photon-photon events is a challenging task, especially at high amounts of pile-up even with an optimistic 10~ps resolution of timing detectors. In contrast, the photon-Pomeron signal is relatively well separable from all backgrounds at low levels of pile-up, allowing us to discover the elastic $t\bar{t}$ production and probe, for the first time, the production of a top quark pair in the photon-Pomeron interactions. The diffractive photoproduction of such a complex system as the $t\bar{t}$ pair hence can be used not only to study diffractive properties of the scattering amplitude but also to search for new physics beyond the Standard Model, and consequently to be a solid part of the physics programme of forward proton detectors at LHC.