The possibility of a theory in which only matter is quantised, and gravity is described by the classical theory of general relativity, even at the fundamental level, is not ruled out by any theoretical argument nor by experiment. Here, I discuss implications of this possibility and show that, from the most naive approach for such a theory, a gravity-matter coupling according to the semiclassical Einstein equations, one obtains a nonlinear Schrödinger equation as a non-relativistic limit that may be used to derive experimental predictions. Possibilities of experimental tests are discussed, which seem feasible with existing technology in the not-too-far future.