The bellows conjecture for small flexible polyhedra in non-Euclidean spaces

The bellows conjecture claims that the volume of any flexible polyhedron of dimension $3$ or higher is constant during the flexion. The bellows conjecture was proved for flexible polyhedra in Euclidean spaces $\mathbb R^n$, $n\ge3$, and for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces $\Lambda^{2m+1}$, $m\ge1$. Counterexamples to the bellows conjecture are known in all open hemispheres $\mathbb S^n_+$, $ n\ge3$. The aim of this paper is to prove that, nonetheless, the bellows conjecture is true for all flexible polyhedra in either $\mathbb S^n$ or $\Lambda^n$, $n\ge3$, with sufficiently small edge lengths.