Uniformization conjecture for bounded domains with parallel Bergman curvature

Let $\Omega \subset \mathbb{C}^n$, $n \geq 1$, be a bounded domain. Write $R^\Omega$ for the curvature tensor of its Bergman metric, and let $A^2(\widetilde\Omega \smallsetminus E)$ denote the Bergman space of square-integrable holomorphic functions on $\widetilde\Omega \smallsetminus E$. Uniformization conjecture. If

$$\nabla R^\Omega = 0,$$

then there exist a bounded symmetric domain $\widetilde\Omega \subset \mathbb{C}^n$ and a subset $E \subset \widetilde\Omega$ of zero Lebesgue measure such that every function in $A^2(\widetilde\Omega \smallsetminus E)$ extends holomorphically to $\widetilde\Omega$, and

$$\Omega \cong \widetilde\Omega \smallsetminus E.$$

The conjecture asks whether the pseudoconvexity hypothesis in the preceding local-symmetry uniformization theorem can be removed. The theorem gives the corresponding conclusion for bounded pseudoconvex locally symmetric domains, while the claim remains open without pseudoconvexity.