Arnold's signature conjecture for domains in real projective space

Consider a domain $U\subset{\mathbb R}P^n$ bounded by a connected smooth hypersurface $B$. Suppose that the second fundamental form of $B$ is non-degenerate at every point and has signature $(n-k-1,k)$, necessarily the same at all points, with respect to the outward normal. Arnold Conjecture. There exist a projective subspace $L^k$ of dimension $k$ contained in $U$ and a projective subspace $L^{n-k-1}$ of dimension $n-k-1$ that does not intersect $U$. This is the signature formulation of Arnold's proposed extension of the convex-body theorem; the paper proves only the first nontrivial case under additional assumptions, leaving the general statement open.