We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between 90◦ and 120◦. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the “tangential” part of the capillary boundary is H^n-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between 90◦ and 180◦. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds. This is joint work with R. Neumayer and R. Resende.