Erez's conjecture on the square root of the inverse different

Let $F_{\pi}/F$ be a finite Galois-algebra extension of number fields with group $G$. Write $\mathfrak{D}(\pi)$ for its different, and suppose that the square root $A(\pi)$ of $\mathfrak{D}(\pi)^{-1}$ is defined. If $F_{\pi}/F$ is weakly ramified, then $A(\pi)$ is a locally free $\mathbf{Z}G$-module and determines a class $(A(\pi))$ in $\operatorname{Cl}(\mathbf{Z}G)$. Let $\Omega(F_{\pi}/F,2)$ denote Chinburg's invariant in this locally free class group. Erez's conjecture. If $F_{\pi}/F$ is weakly ramified, then

$$(A(\pi))=\Omega(F_{\pi}/F,2).$$

This relates the Galois module structure of the square root of the inverse different to Chinburg's equivariant arithmetic invariant. It is known in the tame case, but remains open for general weakly ramified extensions.