  We study scalar curvature deformation for asymptotically locally hyperbolic
(ALH) manifolds with nonempty compact boundary. We show that the scalar
curvature map is locally surjective among either (1) the space of metrics that
coincide exponentially toward the boundary, or (2) the space of metrics with
arbitrarily prescribed nearby Bartnik boundary data. Using those results, we
characterize the ALH manifolds that minimize the Wang-Chru\'sciel-Herzlich mass
integrals in great generality and establish the rigidity of the positive mass
theorems.
