Beaver and Garrity

These authors have made some inroads in Jacobi's problem.

A Two-Dimensional Minkowski ?(x) Function

They write:

"The goal of this section is to show that δ(x, y) maps a class of pairs of cubic irrationals to pairs of rationals. Unfortunately, we cannot make the claim that δ maps all pairs of cubics (even in the same number field) to pairs of rationals."

To this day, one cannot go both ways. Not all pairs of cubics go to pairs of rationals. Only a class of pairs of cubic irrationals goes to pairs of rationals.

These authors end their paper thus:

"Finally, the initial Hermite problem remains open."