Yau's conjecture on Kähler–Einstein Bergman metrics

Let $\Omega$ be a bounded domain that is not a product domain. A domain is homogeneous if its automorphism group acts transitively, meaning that for any $x,y e \Omega$ there is a $g e \operatorname{Aut}(\Omega)$ with $g(x)=y$. Yau's conjecture. The Bergman metric of $\Omega$ is a complete Kähler–Einstein metric if and only if $\Omega$ is homogeneous.

The conjecture concerns when the Bergman metric has the same distinguished curvature property as a Kähler–Einstein metric. The supplied text notes that the unit ball is an example and that the general problem was posed by Yau; no resolution status is given.