While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say about this? Is it fundamentally impossible to derive the Born Rule or is there hope?

Here's a link of great discussion in the same spirit: https://www.math.columbia.edu/~woit/wordpress/?p=10533&cpage=1

In fact here we have someone summarising good reasons/books to believe that it indeed may be possible: http://www.johnboccio.com/research/quantum/notes/bell.pdf

I suspect Lubos Motl was something of the opinion the measurement that should be a separate fundamental axiom. If so, I suspect it becomes a problem of showing: there is no non-redundant axiom when the Born Rule is included? https://motls.blogspot.com/2018/09/woit-and-probability-in-quantum.html