Shafarevich's conjecture on holomorphic convexity of universal covers

Let $X$ be a smooth complex compact Kähler variety, and let $\widetilde{X}$ denote its universal cover. Shafarevich's conjecture. The universal cover $\widetilde{X}$ is holomorphically convex. The conjecture concerns the complex-analytic structure of universal covers of compact Kähler varieties; the source attributes it to Igor R. Shafarevich and gives no resolution status.