The chemical freeze-out of hadrons created in the high energy nuclear collisions is studied within the realistic version of the hadron resonance gas model. The chemical non-equilibrium of strange particles is accounted via the usual $gamma_{s}$ factor which gives us an opportunity to perform a high quality fit with $chi^2/dof simeq 63.5/55 simeq 1.15$ of the hadronic multiplicity ratios measured from the low AGS to the highest RHIC energies. In contrast to previous findings, at low energies we observe the strangeness enhancement instead of a suppression. In addition, the performed $gamma_{s}$ fit allows us to achieve the highest quality of the Strangeness Horn description with $chi^2/dof=3.3/14$. For the first time the top point of the Strangeness Horn is perfectly reproduced, which makes our theoretical horn as sharp as an experimental one. However, the $gamma_{s}$ fit approach does not sizably improve the description of the multi-strange baryons and antibaryons. Therefore, an apparent deviation of multi-strange baryons and antibaryons from chemical equilibrium requires further explanation.
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