The algebraicity conjecture for left-Diophantine numbers

A real number $x\in\mathbb{R}$ is left-Diophantine if it is the supremum of a Diophantine subset of $\mathbb{Q}$. Let $\mathscr{D}$ be the set of left-Diophantine numbers, and let $\mathscr{A}$ be the field of real algebraic numbers. Algebraicity conjecture. All left-Diophantine numbers are algebraic; equivalently, $\mathscr{D}=\mathscr{A}$. This conjecture would give an exact description of the class of left-Diophantine numbers, which is currently known to satisfy $\mathscr{A}\subseteq\mathscr{D}\subseteq\Lambda$, where $\Lambda$ is the class of left-listable real numbers. It follows from the semi-algebraic version of Mazur's conjecture, and hence from the strong version of Mazur's conjecture.