>> Are you doing the "LLMs don't know how many R's are in 'Raspberry'" thing here?
I don't know. Are you doing the thing were you're arguing against something you don't like that the other person never said?
I don't know. Are you doing the thing were you're arguing against something you don't like that the other person never said?
Once you verify (with SymPy or another CAS, like Mathematica) that the given function has the claimed Jacobian determinant (which involves only taking partial derivatives and taking a determinant) and the given inputs map to the same output (which involves only evaluating polynomials with some inputs), you're done.
Besides which you have no idea whether that's how the Jacobian conjecture counterexample was verified. Nobody does because no details were released.