Inspired by the recent TOTEM data for the elastic proton–proton ( pp ) scattering at s = 8 and 13 TeV, we update previous studies of the differential cross sections using the Phillips–Barger (PB) model, which parametrizes the amplitude in terms of a small number of free parameters. We demonstrate that this model is able to describe the recent pp data on a statistically acceptable way. Additionally, we perform separate fits of the pp data for each center-of-mass energy and propose a parametrization for the energy dependence of the parameters present in the PB model. As a consequence, we are able to present the PB predictions for the elastic proton–proton cross section at s = 546 GeV and 1.8 TeV, which are compared with the existing antiproton–proton ( p ¯ p ) data. We show that the PB predictions, constrained by the pp data, are not able to describe the p ¯ p data. In particular, the PB model predicts a dip in the differential cross section that is not present in the p ¯ p data. Such result suggests the contribution of the Odderon exchange at high energies.