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The scale of the changes is too large for a pure replacement of the existing model without more testing.
In the interest of time I therefore put Jan's model as from this PR: #1532 in the sandbox.
I added a version which is the one I have been working on, in an attempt to clean the code a little bit. I will continue to focus on documentation for now, and we can iterate further.
Inversion will come in a separate PR, for now this is purely numerics.