Global hyperbolicity conjecture for cone structures of positive contactomorphism paths

Let $\Sigma$ be a smooth manifold, let $ST^{\ast}\Sigma$ denote its unit cosphere bundle, and let $(f_t)_{t\in\mathbb R}$ be a positive path of contactomorphisms of $ST^{\ast}\Sigma$. Let $C_f$ be the cone structure associated with this path on $\mathbb R\times\Sigma$. Global hyperbolicity conjecture. The cone structure $(\mathbb R\times\Sigma,C_f)$ is globally hyperbolic. The preceding theorem establishes this when the positive path is strongly convex; the conjecture asks whether global hyperbolicity holds for arbitrary positive paths, despite the potentially nonsmooth and complicated boundary of $C_f$.