Gromov's conjecture on convex hypersurfaces in Alexandrov spaces

Let $X\in\operatorname{Alex}^n(\kappa)$ have no boundary, and let a convex hypersurface in $X$ be equipped with its intrinsic metric. Gromov's conjecture. The hypersurface is an Alexandrov space with the same lower curvature bound $\kappa$. This is a long-standing conjecture connecting the intrinsic geometry of convex hypersurfaces with the ambient Alexandrov geometry; the paper mentions it as a motivation for its globalization results, without indicating a resolution.