What do experimental data "say" about growth of hadronic total cross-section?

We reanalyse $\bar{p}p$ and $pp$ high energy data of the elastic scattering above $\sqrt{s}=5$ GeV on the total cross-section ${\sigma }_{tot}$ and on the forward $\rho$-ratio for various models of Pomeron, utilizing two methods. The first one is based on analytic amplitudes, the other one relies on assumptions for ${\sigma }_{tot}$ and on dispersion relation for $\rho$. We argue that it is not possible, from fitting only existing data for forward scattering, to select a definite asymptotic growth with the energy of ${\sigma }_{tot}$. We find equivalent fits to the data together with a logarithmic Pomeron giving a behavior ${\sigma }_{tot}\propto {\ln }^{\gamma }s$, $\gamma \in [0.5,2.20]$ and with a supercritical Pomeron giving a behavior ${\sigma }_{tot}\propto s^{ϵ}$, $ϵ\in [0.01,0.10]$.