  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.
