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

Let $K$ be a number field and let $G$ be a finite abelian group. For a $G$-Galois extension $L/K$, let $\mathcal{A}_{L/K}$ be the associated order and let

$$\operatorname{Tr}_{L/K}(\mathcal{O}_L)^{-1}\mathcal{O}_L$$

be the adjusted ring of integers. Say that $K$ satisfies the modified Leopoldt condition of type $G$ if this adjusted ring of integers is free as an $\mathcal{A}_{L/K}$-module for every $G$-Galois extension $L/K$. mHS-L$(K,G)$ conjecture. If $K$ is a Hilbert-Speiser field of type $G$, then $K$ satisfies the modified Leopoldt condition of type $G$. This modified implication accounts for the trace ideal as a possible global obstruction to freeness over the associated order and is proposed as a natural adjustment of the Leopoldt condition.