The uniqueness conjecture for Leopoldt fields

Let $K$ be a number field. Call $K$ a Leopoldt field if every tame abelian extension $L/K$ has a weak normal integral basis, meaning that, for the maximal order $\mathcal{M}$ of $K[G]$ where $G=\operatorname{Gal}(L/K)$, the module

$$\mathcal{M}\otimes_{\mathscr{O}_K[G]}\mathscr{O}_L$$

is free of rank $1$ over $\mathcal{M}$. Leopoldt-field conjecture. $\mathbb{Q}$ is the only Leopoldt field. This asks whether the uniqueness of $\mathbb{Q}$ among Hilbert–Speiser fields persists for the weaker Leopoldt property; the source does not provide evidence of a resolution.