  Recent results on the slope of the $pp$ diffraction cone measured by TOTEM at
$7$ and $8$ GeV show an unexpected rapid rise in $s$, close to $B(s)\sim
\ln^2$, rather than $\ln s$, typical of the Regge-pole predictions. We show
that the new phenomenon can be accommodated by the inclusion of unitarity
corrections to a simple Regge (pomeron) pole exchange. Interestingly, the
odderon may also promote the acceleration of $B(s)$. The onset of the new
regime may be indicative of the approach to the asymptotic dynamics of strong
interactions. We analyse the new data together with other available forward
measurable in a unitarized Regge dipole. Unitarization proves crucial in
fitting the data, especially those on the slope $B(s)$ showing a change from
the $\ln (s)$ to $\ln^2 (s)$ behavior. Having fitted the free parameters of the
unitarized model to the data, we predict the behavior of the cross sections and
the slope at still higher energies, including those asymptotic.
