  We give examples of spin $4$-manifolds with boundary that do not admit
metrics of positive scalar curvature and nonnegative mean curvature. These
manifolds in fact have the stronger property that the conformal Laplacian with
appropriate boundary conditions is never positive. The obstruction to the
positivity of the conformal Laplacian is given by a real-valued $\xi$-invariant
associated to the APS theorem for the twisted Dirac operator. We use analytic
techniques related to the prescribed scalar curvature problem in conformal
geometry to move beyond earlier work where the metric is a product near the
boundary.
