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.

There are many cases where it's possible to prove that counterexamples must exist, without identifying a specific example. This kind of proof provides more insight into the problem than simply finding a counterexample.