Particle production in relativistic collisions of heavy nuclei is analyzed in a wide range of incident energies 2.7 GeV $le sqrt{s_{NN}}le$ 62.4 GeV. The analysis is performed within the three-fluid model employing three different equations of state (EoS): a purely hadronic EoS, an EoS with the first-order phase transition and that with a smooth crossover transition. It is found that the hadronic scenario fails to reproduce experimental yields of antibaryons (strange and nonstrange), starting already from lower SPS energies, i.e. $sqrt{s_{NN}}>$ 5 GeV. Moreover, at energies above the top SPS one, i.e. $sqrt{s_{NN}}>$ 17.4 GeV, the mid-rapidity densities predicted by the hadronic scenario considerably exceed the available RHIC data on all species. At the same time the deconfinement-transition scenarios reasonably agree (to a various extent) with all the data. The present analysis demonstrates certain advantage of the deconfinement-transition EoSs. However, all scenarios fail to reproduce the strangeness enhancement in the incident energy range near 30A GeV (i.e. a horn anomaly in the $K^+/pi^+$ ratio) and yields of $phi$-mesons at 20A--40A GeV.
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