The minus component of the equivariant Tamagawa number conjecture in determinant form

Let $H/F$ be a finite abelian CM-extension, let $G=\operatorname{Gal}(H/F)$, and let $\Sigma,\Sigma'$ be finite sets of places satisfying the hypotheses used to define the perfect complex $C_{\Sigma}^{\Sigma'}$. Let $\det_{\mathbb{Z}[G]^{-}}(C_{\Sigma}^{\Sigma'})$ denote its determinant module, and let $z^-_{H/F,\Sigma,\Sigma'}$ be the minus-component zeta element obtained from the leading term of the equivariant $L$-function and the regulator isomorphism. The minus eTNC. One has

$$\det_{\mathbb{Z}[G]^{-}}(C_{\Sigma}^{\Sigma'})=z^-_{H/F,\Sigma,\Sigma'}\mathbb{Z}[G]^{-}.$$

Equivalently, $z^-_{H/F,\Sigma,\Sigma'}$ is a basis of the determinant module. The paper states that this formulation is equivalent to the Fitting-ideal formulation above, but the supplied text does not establish that either conjecture is resolved.