I find it interesting that there are some physicists who won't even entertain this question. I remember raising it in a "round table" discussion at a conference and people just laughed and did not want to engage with the question. Hence, it is nice that last year Science published a short piece, Is Quantum Theory Exact? by Stephen Adler and Angelo Bassi.
They discuss a physical collapse model called the continuous spontaneous localization (CSL) model which involves adding noise terms to the Schrodinger equation in order to produce spontaneous wave function collapse on "macroscopic" scales. It should be stressed that this is a radical proposal involving fundamental new "forces" in the universe. They discuss physical bounds from known experiments for the parameters in the model.
Unfortunately, there are several significant issues that the short article does not mention.
Most of the dynamical equations and corresponding experimental signatures that these physical collapse models produce are identical to those for decoherence models. Hence, it will be difficult to experimentally distinguish the CSL model from the ubiquitous effects of decoherence from the environment .
Given that decoherence does not solve the measurement problem because of the problem of definite outcomes the physical collapse models seem to me to do only slightly better, invoking the "gamblers ruin" problem to derive the Born rule.
A broader perspective is given in a 2005 Reviews of Modern Physics article, Decoherence, the Measurement Problem, and the Interpretation of Quantum Mechanics by Max Schlosshauer. He reviews the physical collapse models discussing the above issues. His Summary and Outlook is:
Decoherence has the distinct advantage of being derived directly from the laws of standard quantum mechanics, whereas current collapse models are required to postulate their reduction mechanism as a new fundamental law of nature. On the other hand, collapse models yield, at least for all practical purposes, proper mixtures, so they are capable of providing an “objective” solution to the measurement problem. The formal similarity between the time evolution equations of the collapse and decoherence models nourishes hopes that the postulated reduction mechanisms of collapse models could possibly be derived from the ubiquituous and inevitable interaction of every physical system with its environment and the resulting decoherence effects. We may therefore regard collapse models and decoherence not as mutually exclusive alternatives for a solution to the measurement problem, but rather as potential candidates for a fruitful unification.