The non-Diophantineness conjecture for the integers over the rationals

Let $\mathbb{Z}$ denote the ring of integers and $\mathbb{Q}$ the field of rational numbers. A subset of $\mathbb{Q}$ is Diophantine if it is defined by the existence of solutions to a polynomial equation over $\mathbb{Q}$. The non-Diophantineness conjecture. There is no Diophantine definition of $\mathbb{Z}$ over $\mathbb{Q}$. This conjecture would rule out transferring the negative solution of Hilbert's tenth problem over $\mathbb{Z}$ by a Diophantine definition of the integers in the rationals; it is stated as Mazur's Second Conjecture and remains unresolved.