  We make use of the powerful formalism of quantum parameter estimation to
assess the characteristic rates of a Continuous Spontaneous Localisation (CSL)
model affecting the motion of a massive mechanical system. We show that a study
performed in non-equilibrium conditions unveils the advantages provided by the
use of genuinely quantum resources -- such as quantum correlations -- in
estimating the CSL-induced diffusion rate. In stationary conditions, instead,
the gap between quantum performance and a classical scheme disappears. Our
investigation contributes to the ongoing effort aimed at identifying suitable
conditions for the experimental assessment of collapse models.
