  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.
