Very precise data on elastic proton-proton scattering at $sqrt{s}=7$, $8$ and $13$ TeV have been obtained by the TOTEM group at LHC in the near-forward region (momentum transfers down to $|t| = 6 times 10^{-4} {rm GeV}^2$ at $sqrt{s}=8$ TeV and to $|t| = 8 times 10^{-4} {rm GeV}^2$ at $sqrt{s}=13$ TeV). The Coulomb-nuclear interference has been measured with sufficient accuracy for TOTEM to establish the falloff of the $rho$ parameter with increasing energy. The predictions from a previously studied model are shown to be in good agreement with the data and thus allow us to draw rather firm conclusions about the structure of the near-forward nuclear amplitude. We point out that due to a zero in the real part of the nuclear amplitude occurring at a very small momentum transfer--that can migrate into the Coulomb-nuclear interference (CNI) region at higher energies--much care is needed in extracting the numerical value of $rho$ for such energies. Thus, the true value of $rho$ would be higher than the TOTEM value for $rho$ found under the hypothesis that the real part of the elastic nuclear amplitude is devoid of such a zero in the CNI region.
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