Models when predictors and group effects correlate

With regards to a standard hierarchical linear regression model, do we need to account for predictor and group effect correlations as described in Bafumi and Gelman (2007), or is this satisfied by using a multivariate normal with LKJ distribution for the covariance matrix (using the Cholesky decomposition parameterization)? Any clarification on the differences in these approaches, both theoretical and practical, is appreciated.

Bafumi, Joseph, and Andrew Gelman. 2007. “Fitting Multilevel Models When Predictors and Group Effects Correlate.” SSRN Electron. J. Fitting Multilevel Models When Predictors and Group Effects Correlate by Joseph Bafumi, Andrew Gelman :: SSRN.