  We prove a Gannon-Lee theorem for strongly causal and null pseudoconvex
Lorentzian metrics of regularity $C^1$. Thereby we generalize earlier results
to the non-globally hyperbolic setting in a natural way, while at the same time
lowering the regularity to the most general class currently available in the
context of the classical singularity theorems. Along the way we also prove that
any maximizing causal curve in a $C^1$-spacetime is a geodesic and hence of
$C^2$-regularity.
