  Roger Penrose proposed that a spatial quantum superposition collapses as a
back-reaction from spacetime, which is curved in different ways by each branch
of the superposition. In this sense, one speaks of gravity-related wave
function collapse. He also provided a heuristic formula to compute the decay
time of the superposition $\mathord{-}$ similar to that suggested earlier by
Lajos Di\'osi, hence the name Di\'osi-Penrose model. The collapse depends on
the effective size of the mass density of particles in the superposition, and
is random: this randomness shows up as a diffusion of the particles' motion,
resulting, if charged, in the emission of radiation. Here, we compute the
radiation emission rate, which is faint but detectable. We then report the
results of a dedicated experiment at the Gran Sasso underground laboratory to
measure this radiation emission rate. Our result sets a lower bound on the
effective size of the mass density of nuclei, which is about three orders of
magnitude larger than previous bounds. This rules out the natural
parameter-free version of the Di\'osi-Penrose model.
