  The possibility that the collapse of the wave function in quantum mechanics
is a real and ultimately connected to (classical) gravity has been debated for
decades, with main contributions by Di\'osi and Penrose. In particular, Di\'osi
proposed a noise-based dynamical reduction model, which captures the same
orders of magnitude for the collapse time suggested by Penrose based on
heuristic arguments. This is known in literature as the DP model
(Di\'osi-Penrose). A peculiarity of the DP model is the prediction of
spontaneous heating of matter, which can be tested without the need for massive
quantum superpositions. Notably, a very similar effect is predicted by recent
theoretical approaches to gravity as a classical-only information channel.
Here, we reconsider the current constraints on the DP model from spontaneous
heating, by analyzing experimental situations not properly considered before.
We argue that the the parameter-free version of the DP model is close to be
ruled out by standard heat leak measurements at ultralow temperature, with a
conclusive exclusion likely within reach with existing technology. This result
would strengthen a recent claim of exclusion inferred by spontaneous x-ray
emission experiments, which relies on the somewhat stronger assumption that the
DP noise field is white up to x-ray frequencies.
