  We present a model-independent analysis of the experimental data on the ratio
$X$ between the elastic and total cross-sections from $pp$ and $\bar{p}p$
scattering in the c.m. energy interval 5 GeV - 8 TeV. Using a novel empirical
parametrization for that ratio as a function of the energy and based on
theoretical and empirical arguments, we investigate three distinct asymptotic
scenarios: either the black-disk (BD) limit or scenarios above and below that
limit. Our analysis favors a scenario below the BD, with asymptotic ratio $X =
0.36 \pm 0.08$.
