The symplectic-structure conjecture for globally hyperbolic four-manifolds

Let $(M,\mathrm{g})$ be an orientable $4$-dimensional globally hyperbolic Lorentzian manifold with an orthogonal decomposition of its tangent bundle $TM=N_\star\oplus N$, where $N$ is a trivial normal bundle and $N_\star$ is tangent to a symplectic foliation. Let $\nabla^\star$ and $\nabla^N$ be the specified hermitian connexions on the corresponding complex line bundles, and let $R$ be the vector field used in the construction. Symplectic-structure conjecture. If $N_\star$ is not a trivial hermitian line bundle and $\nu$ is the proposed potential $1$-form for $\nabla^N$, then

$$\varpi:=\sqrt{-1}\operatorname{curv}(\nabla^\star)+\sqrt{-1}\operatorname{curv}(\nabla^N)$$

is a symplectic structure wherever $R$ does not vanish. The claim extends the verified Schwarzschild case to the stated class of globally hyperbolic spacetimes, but the source provides no resolution in general.