Recently, there have been several evidences that the hadronic total cross section ${\ensuremath{\sigma}}_{\mathrm{tot}}$ is proportional to $B{log}^{2}s$, which is consistent with the Froissart unitarity bound. The COMPETE Collaboration has further assumed ${\ensuremath{\sigma}}_{\mathrm{tot}}\ensuremath{\simeq}B{log}^{2}(s/{s}_{0})+Z$ to extend its universal rise with the common values of $B$ and ${s}_{0}$ for all hadronic scatterings to reduce the number of adjustable parameters. It was suggested that the coefficient $B$ was universal in the arguments of the color glass condensate of QCD in recent years. However, there has been no rigorous proof yet based only on QCD. We attempt to investigate the value of $B$ for ${\ensuremath{\pi}}^{\ensuremath{\mp}}p$, ${K}^{\ensuremath{\mp}}p$ and $\overline{p}p$, $pp$ scatterings, respectively, through the search for the simultaneous best fit to the experimental ${\ensuremath{\sigma}}_{\mathrm{tot}}$ and $\ensuremath{\rho}$ ratios at high energies. The ${\ensuremath{\sigma}}_{\mathrm{tot}}$ at the resonance- and intermediate-energy regions has also been exploited as a duality constraint based on the special form of the finite-energy sum rule. We estimate the values of $B$, ${s}_{0}$, and $Z$ individually for ${\ensuremath{\pi}}^{\ensuremath{\mp}}p$, ${K}^{\ensuremath{\mp}}p$ and $\overline{p}p$, $pp$ scatterings without using the universality hypothesis. It turns out that the values of $B$ are mutually consistent within 1 standard deviation. It has to be stressed that we cannot obtain such a definite conclusion without the duality constraint. It is also interesting to note that the values of $Z$ for $\ensuremath{\pi}p$, $Kp$, and $\overline{p}(p)p$ approximately satisfy the ratio $2\ensuremath{\mathbin:}2\ensuremath{\mathbin:}3$ predicted by the quark model. The obtained value of $B$ for $\overline{p}(p)p$ is ${B}_{pp}=0.280\ifmmode\pm\else\textpm\fi{}0.015\text{ }\text{ }\mathrm{mb}$, which predicts ${\ensuremath{\sigma}}_{\mathrm{tot}}^{pp}=108.0\ifmmode\pm\else\textpm\fi{}1.9\text{ }\text{ }\mathrm{mb}$ and ${\ensuremath{\rho}}^{pp}=0.131\ifmmode\pm\else\textpm\fi{}0.0025$ at the LHC energy $\sqrt{s}=14\text{ }\text{ }\mathrm{TeV}$.