Before revision:  The previous approach relied heavily on linear fixed effect specifications that severely limit the heterogeneity between clusters. These methods imply that linearly adjusting for differences between clusters in average covariate values addresses all concerns with cross-cluster comparisons.
After revision:  The previous approach relied heavily on linear fixed effect specifications that severely limit the extent of the heterogeneity between groups by making the restrictive assumption that linearly adjusting for differences between clusters in average covariate values addresses all concerns with cross-cluster comparisons.