Arnold's hypothesis on hyperbolic hypersurfaces

A smooth hypersurface in $\mathbb{RP}^{n+1}$ is $(k,l)$-hyperbolic if its second quadratic form has signature $(k,l)$. Let $U\subset\mathbb{R}P^n$ be a domain bounded by a connected smooth $(k,l)$-hyperbolic hypersurface $B$. Arnold's hypothesis. There exist a projective subspace $L_-$ of dimension $k$ not intersecting $U$ and a projective subspace $L_+$ of dimension $l$ contained in $U$. Any projective line joining a point of $L_+$ and a point of $L_-$ intersects $B$ at exactly two points. The claim generalizes the corresponding property of quadrics and concerns the existence of complementary projective subspaces controlled by the signature of the second quadratic form. Its resolution status is not specified in the source.