Vinatier's freeness conjecture for the square root of the inverse different

Let $L/K$ be a finite weakly ramified Galois extension of number fields, with $G=\operatorname{Gal}(L/K)$, ring of integers $\mathcal{O}_L$, and different $\mathcal{D}_{L/K}$. When it exists, let $\mathcal{A}_{L/K}$ be the unique fractional ideal satisfying

$$(\mathcal{A}_{L/K})^2=(\mathcal{D}_{L/K})^{-1}.$$

For an extension of odd degree, this ideal exists and is a $\mathbb{Z}[G]$-module. Vinatier's conjecture. If $L/K$ is a weakly ramified Galois extension of number fields of odd degree, then $\mathcal{A}_{L/K}$ is a free $\mathbb{Z}[G]$-module. The conjecture extends the known result for tamely ramified extensions, where the square root of the inverse different is free over the integral group ring. Its status is not resolved in the supplied source.