The key debate, around the measurement problem is whether collapse should be interpreted as a physical process(Bohmian Mechanics) or as an immaterial process(e.g. Copenhagen Interpretation, Consciousness causes collapse, etc.). It has been confirmed that wave function collapse happens instantaneously. I think one way to resolve this issue could be to construct a mathematical theorem which demonstrates that physical causality always takes time to occur or at a finite speed. By determining that it is impossible for physical causation to take place instantaneously, one could then infer that the process of wave function collapse is non physical and one can thus eliminate interpretations with a physical mechanism for collapse. Do you think that it is possible to construct such a mathematical theorem?
One does need to specify a bit more about what one means by the Copenhagen interpretation. There is the form involving Bohr's notion of complementarity, in which some form of physical reality is ascribed to wave functions (this is so vaguely expressed that might even include Bohmian mechanics), and there is what Jeffrey Bub calls the the orthodox interpretation, or Dirac-Von Neumann interpretation, in which the wave function is only an expression of probability, and, like any probability, exists only in the mind of the observer.
I think one way to resolve this issue could be to construct a mathematical theorem which demonstrates that physical causality always takes time to occur or at a finite speed.
This theorem already exists in quantum electrodynamics (and by implication in quantum mechanics generally). It is the locality, or microcausality condition, and it is necessary to obtain unambiguous probabilities.
There are other approaches too. It is an undeniable principle that in many repetitions of an experiment, we should expect a certain relative frequencies for each possible result. From this one can construct the whole mathematical structure of quantum mechanics as a probability theory for indeterminate processes, including the principle of superposition and the derivation of the Schrodinger equation from which apparent interference effects arise. This shows that there is nothing more mysterious to wave function collapse than the change to probability when information becomes known.
In my view, this means that the measurement problem has already been resolved by mathematical theorems; we already have a unique consistent interpretation. The continuing argument is not resolved by mathematics any more than the deniers of relativity recognise Einstein's mathematical arguments, and is not different in principle than e.g. argument over Dingle's fallacies. If there is a difference it is that the mathematical foundations of quantum mechanics involves more advanced mathematics and is accessible to fewer people.
I have given a complete treatment to clarify the orthodox interpretation in The Hilbert space of conditional clauses