According to the following paper:

Carlos R. Videla, On the constructible numbers, Proceedings of the American Mathematical Society Vol. 127, No. 3 (Mar., 1999), pp. 851-860.

The problem has remained open at least until 1999. I think the problem is still open. In the above paper the author proves that the ring of constructible algebraic integers is first-order definable in the field of constructible numbers. The author hopes that $\mathbb{Z}$ should be definable in the ring of constructible algebraic integers and therefore his result would be a partial result towards resolving the problem negatively.

According to the following paper:

• Carlos R. Videla, On the constructible numbers, Proceedings of the American Mathematical Society Vol. 127, No. 3 (Mar., 1999), pp. 851-860. https://doi.org/10.1090/S0002-9939-99-04611-0

The problem has remained open at least until 1999. I think the problem is still open. In the above paper the author proves that the ring of constructible algebraic integers is first-order definable in the field of constructible numbers. The author hopes that $\mathbb{Z}$ should be definable in the ring of constructible algebraic integers and therefore his result would be a partial result towards resolving the problem negatively.