  Recently the TOTEM experiment at the LHC has released measurements at
$\sqrt{s} = 13$ TeV of the proton-proton total cross section, $\sigma_{tot}$,
and the ratio of the real to imaginary parts of the forward elastic amplitude,
$\rho$. Since then an intense debate on the $C$-parity asymptotic nature of the
scattering amplitude was initiated. We examine the proton-proton and the
antiproton-proton forward data above 10 GeV in the context of an eikonal
QCD-based model, where nonperturbative effects are readily included via a QCD
effective charge. We show that, despite an overall satisfactory description of
the forward data is obtained by a model in which the scattering amplitude is
dominated by only crossing-even elastic terms, there is evidence that the
introduction of a crossing-odd term may improve the agreement with the
measurements of $\rho$ at $\sqrt{s} = 13$ TeV. In the Regge language the
dominant even(odd)-under-crossing object is the so called Pomeron (Odderon).
