The weak separation conjecture for left-listable and left-Diophantine numbers

A real number $\alpha$ is left-listable if $L(\alpha)=\{q\in\mathbb{Q}:q<\alpha\}$ is listable. Let $\mathscr{D}$ be the set of left-Diophantine numbers and let $\Lambda$ be the set of left-listable numbers. Weak separation conjecture. Not every left-listable number is left-Diophantine; equivalently, $\mathscr{D}\ne\Lambda$. This is weaker than the algebraicity conjecture and would already imply that $\mathbb{Z}$ is not Diophantine in $\mathbb{Q}$, that $\mathbb{Q}$ does not have the DPRM property, and that $\mathbb{Q}$ and $\mathbb{N}$ are not positive existentially bi-interpretable.