  We prove that any weakly symplectically fillable contact manifold is tight.
Furthermore we verify the strong Weinstein conjecture for contact manifolds
that appear as the concave boundary of a directed symplectic cobordism whose
positive boundary satisfies the weak-filling condition and is overtwisted.
Similar results are obtained in the presence of bordered Legendrian open books
whose binding-complement has vanishing second Stiefel-Whitney class. The
results are obtained via polyfolds.
