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 factor which accounts for the remaining strange particle non-equilibration. Within suggested approach the parameters of two chemical freezeouts 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 2 / ≃ 0.93. A special attention is paid to a successful description of the Strangeness Horn. The well-known problem of selective suppression ofΛ andΞ hyperons is also discussed. The main result is that for all collision energies the 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.