The Zubarev approach of the non-equilibrium statistical operator is used to account for the enhancement of the low-$p_T$ part of pion spectra by introducing an effective pion chemical potential. This is an alternative to the explanation of the low-$p_T$ enhancement by resonance decays. We report on the first results obtained with a newly developed thermal particle generator that implements both mechanisms of low-$p_T$ enhancement and applies Bayesian inference methods for these scenarios to find the most probable sets of thermodynamic parameters at the freeze-out hypersurface for the case of the transverse momentum spectra of identified particles measured by the ALICE Collaboration.