$\DeclareMathOperator\Conf{Conf}\DeclareMathOperator\Iso{Iso}$Let $ M $ be a locally homogeneous Riemannian manifold, in other words the universal cover $ \tilde{M} $ has a transitive action by the isometry group $ \Iso(\tilde{M}) $. Let $ \Iso(M) $ be the group of isometries of $ M $ and let $ \Conf(M) $ be the conformal group of $ M $ (the group of all diffeomorphisms of $ M $ that preserve the conformal structure induced by the metric).
It is well known that the flat space $ E^n $ and the round sphere $ S^n $ have conformal groups strictly larger than their isometry groups since $ \Iso(E^n)=\mathbb{R}^n \rtimes O(n) $ and $ \Iso(S^n)=O(n+1) $ while the full conformal group includes spherical inversions and so is enlarged to include an indefinite orthogonal group.
Besides these two exceptions, in other words excluding all $ M $ with universal cover $ E^n $ or $ S^n $, is it true that any locally homogeneous manifold has $ \Conf(M)=\Iso(M) $?
