The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special symmetries.
When it comes to higher dimensional hyperbolic spaces then the isometry group still contains non-uniform, non-arithmetic lattices. Therefore I wonder what is known or expected about the existence or non-existence of Maass cusp forms for these lattices. For example, is there is an analogon of the Phillips-Sarnak conjecture?