Diophantine undecidability conjecture for subfields of the maximal abelian extension

Let $\mathbb{Q}^{\operatorname{ab}}$ denote the maximal abelian extension of $\mathbb{Q}$, and let $\mathcal{O}_{\mathbf L}$ denote the ring of integers of a subfield $\mathbf L$. Diophantine-undecidability conjecture. For each prime $p\in\{7,11,13\}$, there are subfields $\mathbf L\subset\mathbb{Q}^{\operatorname{ab}}$ such that $\mathbb{Q}^{\operatorname{ab}}/\mathbf L$ is cyclic of degree $p$ and $\mathcal{O}_{\mathbf L}$ is diophantine undecidable. This is a proposed source of examples of undecidable rings of integers in large abelian extensions; the source provides no resolution.