Local homogeneity implies Bergman logarithmic flatness

Let $(M, g; L, h)$ be a polarized manifold, and let $S$ be the circle bundle of $(L^*, h^*)$. The manifold $(M,g)$ is locally homogeneous if its local biholomorphic isometries act transitively. Local homogeneity conjecture. If $(M, g)$ is locally homogeneous, then $S$ is Bergman logarithmically flat. This conjecture asks whether the compactness assumption in the preceding criterion can be removed. The authors propose to study it in the non-compact manifold case in a separate article.