  In a recent paper O. Gannot and M. Wrochna considered the Klein-Gordon
equation on an asymptotically anti-de Sitter spacetime subject to Robin
boundary conditions, proving in particular a propagation of singularity
theorem. In this work we generalize their result considering a more general
class of boundary conditions implemented on the conformal boundary via
pseudodifferential operators of suitable order. Using techniques proper of
$b$-calculus and of twisted Sobolev spaces, we prove also for the case in hand
a propagation of singularity theorem along generalized broken
bicharacteristics, highlighting the potential presence of a contribution due to
the pseudodifferential operator encoding the boundary condition.
