Cheng's conjecture on Kähler–Einstein Bergman metrics

Let $\Omega\subset\mathbb{C}^n$ be a bounded strictly pseudoconvex domain with $C^\infty$ boundary, and let $\omega_B$ denote its Bergman metric. Cheng's conjecture. If $\omega_B$ is Kähler–Einstein, then $\Omega$ is biholomorphic to the ball.

This is presented as a more tractable strictly pseudoconvex, smooth-boundary case of Yau's conjecture. The supplied text gives no evidence that it has been solved or refuted.