Chinburg's $\Omega(3)$ conjecture for number-field extensions

Let $F/K$ be a finite Galois extension of number fields with Galois group $G$. Choose a finite $G$-stable set $S$ of places satisfying the stated hypotheses, let $X_S$ be the kernel of the augmentation map $\Z[S]\to\Z$, and let a Tate sequence representing the canonical class be

$$0\longrightarrow\mathcal{O}_{F,S}^{\times}\longrightarrow A\longrightarrow B\longrightarrow X_S\longrightarrow0.$$

Define $\Omega(F/K,3)=[A]-[B]\in\K_0(\Z[G])$, viewed in the locally free class group $\Cl(\Z[G])$, and let $W_{F/K}\in\Cl(\Z[G])$ be the root number class.

Chinburg's $\Omega(3)$ conjecture.

$$\Omega(F/K,3)=W_{F/K}.$$

This conjecture predicts that the class obtained from the Tate sequence equals the class determined by Artin root numbers. The supplied text records the conjecture and notes that the root number class has order at most two, but gives no general resolution.