The eTNC minus component in Fitting-ideal form for CM-extensions

Let $H/F$ be a finite abelian CM-extension, and write $G=\operatorname{Gal}(H/F)$. Let $\Sigma$ and $\Sigma'$ be finite sets of places of $F$ satisfying (H1), (H2), (H3), and (H4). Let $\Omega_{\Sigma}^{\Sigma'}$ be the associated $\mathbb{Z}[G]^{-}$-module and let $\theta_{\Sigma}^{\Sigma'}$ be the associated Stickelberger element. The eTNC$^{-}$. One has

$$\operatorname{Fitt}_{\mathbb{Z}[G]^{-}}(\Omega_{\Sigma}^{\Sigma'})=(\theta_{\Sigma}^{\Sigma'})$$

as ideals of $\mathbb{Z}[G]^{-}$. This is the main equivariant Tamagawa number conjecture considered in the paper; its status is not resolved by the supplied text.