I will discuss basic ideas and some benefits of the conformally flat inhomogeneous pressure Stephani models of the universe. Their spherically symmetric version is compementary to inhomogeneous density Lemaitre-Tolman-Bondi (LTB) models. I will present the results of checking these models against supernovae data only for off-centers observers as well as for all the data (supernovae, BAO, CMB) for the centrally-placed observers. The results show that similarly as for LTB models, in order to fit the data, the inhomogeneity has to be small, though still non-vanishing.
Finally, I will comment on the possible advantage (conformal flatness accompanied by inhomogeneity) of these models in view of the Green and Wald theorem about backreaction effect in cosmology.