  In this note, we establish the dihedral rigidity phenomenon for a collection
of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds,
extending Gromov's comparison theory to metrics with negative scalar curvature
lower bounds. Our result is a localization of the positive mass theorem for
asymptotically hyperbolic manifolds.
  We also motivate and formulate some open questions concerning related
rigidity phenomenon and convergence of metrics with scalar curvature lower
bounds.
