Self-intersection conjecture for flexible cross-polytopes in Euclidean and Lobachevsky spaces

Let $n\ge 3$. A flexible cross-polytope is a flexible polyhedral cross-polytope in the indicated ambient space. Self-intersection conjecture. All flexible cross-polytopes in $\mathbb{R}^n$ and in $\Lambda^n$ are self-intersecting. The corresponding assertion is false for spheres and open hemispheres, where embedded flexible cross-polytopes exist; the conjecture concerns the Euclidean and Lobachevsky cases.