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.
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