Martin Davis's conjecture on simple sets and Hilbert's tenth problem over the rationals

Let $S\subseteq\mathbb{N}$ be a simple set, meaning that $\mathbb{N}\setminus S$ is infinite and contains no infinite recursively enumerable subset. Let $U\subseteq\mathbb{Q}$ be the set used in the preceding construction, and define

$$S_p=\{a\in\mathbb{N}:\exists x\in\mathbb{Q}^m\text{ such that }p(y_a,x)=0\}.$$

Here $p$ is the polynomial associated with a Diophantine definition of $S$ and the map $a\mapsto y_a$ from the preceding construction. Martin Davis's conjecture. There is a Diophantine definition of a simple set $S$ for which $\mathbb{N}\setminus S_p$ is infinite. If true, this would imply that Hilbert's tenth problem is unsolvable over $\mathbb{Q}$; the source provides no resolution status.