  We present an elaborate version of the hadron resonance gas model with the
combined treatment of separate chemical freeze-outs for strange and non-strange
hadrons and with an additional $\gamma_{s}$ factor which accounts for the
remaining strange particle non-equilibration. Within suggested approach the
parameters of two chemical freeze-outs are connected by the conservation laws
of entropy, baryonic charge, third isospin projection and strangeness. The
developed model enables us to perform a high-quality fit of the hadron
multiplicity ratios measured at AGS, SPS and RHIC with $\chi^2/dof \simeq
0.93$. A special attention is paid to a successful description of the
Strangeness Horn. The well-known problem of selective suppression of $\bar
\Lambda $ and $\bar \Xi$ hyperons is also discussed. The main result is that
for all collision energies the $\gamma_{s}$ factor is about 1 within the error
bars, except for the center of mass collision energy 7.6 GeV at which we find
about 20\% enhancement of strangeness. Also we confirm an existence of strong
jumps in pressure, temperature and effective number of degrees of freedom at
the stage of strange particle chemical freeze-out, when the center of mass
collision energy changes from 4.3 to 4.9 GeV. We argue that these
irregularities may signal about the quark-hadron phase transition.
