The bellows conjecture for non-Euclidean spaces

Let $n\ge 3$, and let a flexible polyhedron lie either in Lobachevsky space $\Lambda^n$ or in the open hemisphere $S^n_+$. Its generalized oriented volume is defined in the relevant ambient space; in the spherical case it is understood modulo the volume group $\mathbb{R}/\sigma_n\mathbb{Z}$, where $\sigma_n$ is the volume of $S^n$. The bellows conjecture for non-Euclidean spaces. The generalized oriented volume of any such flexible polyhedron is constant during the flexion. The assertion is formulated to avoid the known spherical counterexamples involving polyhedra containing antipodal points; the supplied text does not state whether the conjecture has been resolved.