The HS-L conjecture for Hilbert-Speiser and Leopoldt fields of type $G$

Let $K$ be a number field and let $G$ be a finite abelian group. A number field $K$ is a Hilbert-Speiser field of type $G$ if, for every tame $G$-Galois extension $L/K$, the ring of integers $\mathcal{O}_L$ is free as an $\mathcal{O}_K[G]$-module. It is a Leopoldt field of type $G$ if, for every $G$-Galois extension $L/K$, the ring of integers $\mathcal{O}_L$ is free as an $\mathcal{A}_{L/K}$-module, where

$$\mathcal{A}_{L/K}=\{x\in K[G]:x(\mathcal{O}_L)\subseteq\mathcal{O}_L\}.$$

HS-L$(K,G)$ conjecture. $K$ is a Hilbert-Speiser field of type $G$ if and only if $K$ is a Leopoldt field of type $G$. This generalizes the equivalence of the unrestricted Hilbert-Speiser and Leopoldt properties; it is known when $G=C_p$ has prime order $p$ and $K$ contains a primitive $p$th root of unity, while the general case remains open.