The converse déjà vu conjecture for skies of causally related points

Let $\mathcal{X}$ be a globally hyperbolic spacetime whose Cauchy surface is not homeomorphic to the sphere. Let $x,y\in\mathcal{X}$ be causally related points, and consider future-directed timelike curves connecting $x$ to $y$. A converse déjà vu conjecture asserts that if every such curve has déjà vu moments, then the skies of $x$ and $y$ either intersect or form a déjà vu Legendrian link in $\mathfrak{N}_{\mathcal{X}}$. This conjecture proposes a converse to the preceding implication for spacetimes with nonspherical Cauchy surface; its status is not resolved in the supplied source context.