I would frame it differently. The existence of compact counterexamples to a true-seeming conjecture suggests that there’s some deeper understanding waiting to be discovered. Fuzz testing for theorems, if that makes sense. I hope mathematicians in 2036 will be able to explain in detail why the Jacobian conjecture was false and identify which similar, true conjectures the community’s intuition was pointing towards.

We can take a simpler example. Let's say someone conjectures that all linear maps are isomorphic if they have the same domain and codomain*. A counterexample is easy to find, but true insight would be to notice that all linear maps with the same domain and codomain that are not isomorphic map some non-zero elements to zero. That is much more interesting than just finding a counterexample. Although, that isn't to say that finding a counterexample is not very interesting.

*statements only apply to maps whose domain is finite-dimensional

I suspect an AI, possibly a successor to current LLMs, will achieve that by the early 2030s. It might help illuminate many mysteries in math and beyond for us all.