The field conjecture for left-Diophantine numbers

Let $\mathscr{D}\subseteq\mathbb{R}$ be the set of left-Diophantine numbers. Field conjecture. The set $\mathscr{D}$ is a field. This is a weaker consequence of the algebraicity conjecture, since the real algebraic numbers form a field. It would distinguish $\mathscr{D}$ from the larger class $\Lambda$ of left-listable numbers, which is not even a ring.