  We investigate the Di\'osi-Penrose (DP) proposal for connecting the collapse
of the wave function to gravity. The DP model needs a free parameter, acting as
a cut-off to regularize the dynamics, and the predictions of the model highly
depend on the value of this cut-off. The Compton wavelength of a nucleon seems
to be the most reasonable cut-off value since it justifies the non-relativistic
approach. However, with this value, the DP model predicts an unrealistic high
rate of energy increase. Thus, one either is forced to choose a much larger
cut-off, which is not physically justified and totally arbitrary, or one needs
to include dissipative effects in order to tame the energy increase. Taking the
analogy with dissipative collisional decoherence seriously, we develop a
dissipative generalization of the DP model. We show that even with dissipative
effects, the DP model contradicts known physical facts, unless either the
cut-off is kept artificially large, or one limits the applicability of the
model to massive systems. We also provide an estimation for the mass range of
this applicability.
