The bidisk conclusion for domains without totally real boundary points

Let $M$ be a hyperbolic complex surface, and let $\Omega\subset\subset M$ be a subdomain with $C^{1,1}$ boundary. A compact quotient of $\Omega$ means that $\Omega$ admits a quotient by a group of biholomorphisms that is compact. A boundary point is totally real in the sense used for the boundary of $\Omega$. Bidisk conjecture. If $\Omega$ admits a compact quotient and $\partial\Omega$ does not contain a totally real boundary point, then the universal cover of $\Omega$ is biholomorphic to a bidisk. This is presented as a potential extension of the result of Cheung et al. that, under a smooth-boundary hypothesis, the universal cover is either a ball or a bidisk. The source replaces smoothness by the weaker $C^{1,1}$ condition; the resolution status of this asserted consequence is not specified in the supplied text.