Stability of global hyperbolicity for cone-bundle sections

Let $F$ be the bundle whose sections define the relevant causal structures, and let $\mathrm{\Gamma}(F)$ be its space of sections with the Whitney fine $C^0$ topology. Write $\mathrm{\Gamma}_H(F)$ for the globally hyperbolic sections, and $\mathrm{\Gamma}(F,C)$ for sections strictly slower than a cone bundle $C$. Stability of global hyperbolicity. The set $\mathrm{\Gamma}_H(F)$ is open in $\mathrm{\Gamma}(F)$, and

$$\mathrm{\Gamma}_H(F)=\bigcup_{\text{globally hyperbolic }C}\mathrm{\Gamma}(F,C).$$

Global hyperbolicity is known to be stable for Lorentzian cone bundles, while the corresponding assertion in the general cone-bundle setting remains to be proved; the conjecture supplies the technical stability needed for a direct cover by globally hyperbolic cone bundles.