Local homogeneity conjecture for holomorphic Riemannian metrics

Let $X$ be a compact complex manifold bearing a holomorphic Riemannian metric.

Local homogeneity conjecture. Any holomorphic Riemannian metric on $X$ is locally homogeneous.

This conjecture is motivated by the known result that holomorphic Riemannian metrics on compact complex threefolds are locally homogeneous and, after a finite unramified cover, have constant sectional curvature. Its status is not resolved in the supplied source.