From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact convex domain) in the $3$-space form of constant curvature $\kappa$.

For $\kappa=0$, the surface is unique up to isometry of the surrounding space $\mathbb{R}^3$ due to Pogorelov. Is the same true for the elliptic and hyperbolic case? In other words, are two closed convex surfaces in these spaces congruent when they are isometric with respect to their inner metrics?

I asked this question here on Math SE, but did not get an answer.