Revealing self-gravity in a Stern-Gerlach Humpty-Dumpty experiment

There is no consensus among today's physicists about how to describe the gravitational interaction properly in a quantum framework. We propose in this paper an experimental test aimed at revealing the existence of a non-linear self-interaction \`a la Schrodinger-Newton (S-N). In this test, a mesoscopic spin 1/2 microsphere is freely falling in a Humpty-Dumpty Stern-Gerlach interferometer. After clarifying the role of the scaling of the interaction in function of the amplitudes of the up and down spin components of the microsphere, it is shown that self-gravity induces a measurable phase shift between them, which paves the way to experimental tests. It is also shown that if we consider two distinct microspheres falling in parallel, the entangling power of the S-N interaction is exactly equal to zero.