The semialgebraic universal-cover classification conjecture

Let $X$ be a normal, projective variety over ${\mathbb C}$ with universal cover $\tilde X$. A semialgebraic open subset of a projective variety means an open subset that is semialgebraic in a projective embedding. A bounded symmetric domain is a bounded symmetric domain in a complex affine space.

Semialgebraic universal-cover classification conjecture. The following are equivalent:

$$\tilde X \text{ is biholomorphic to a semialgebraic open subset of a projective variety.}$$

and

$$\tilde X \text{ is biholomorphic to } {\mathbb D}\times {\mathbb C}^m\times F,$$

where ${\mathbb D}$ is a bounded symmetric domain, $m\geq 0$, and $F$ is a normal, projective, simply connected variety.

This extends the classification of varieties with quasi-projective universal cover to the broader semialgebraic setting, including compact quotients of bounded symmetric domains. The statement is presented as the proposed classification for normal projective varieties; the supplied text gives no resolution, so its status remains open.