The assumptions underlying the elliptic relaxation model for the pressure term in the Reynolds stress transport equations are examined through the analysis of a channel flow DNS database. The model is shown to be consistent with the data, in particular with regard to the evaluation of the length scale. Some features not accounted by the model, such as the asymmetry in the inhomogeneous direction of the two-point correlation function, are found to be responsible for the observed spurious amplification of the return to isotropy in the log layer. The expected reduction is obtained in the proposed new formulations of the elliptic relaxation equation. The common belief that this reduction is due to the wall echo effect is shown to be erroneous.