Kurihara's determinant conjecture for minus theta-kernel bases

Let $H/F$ be a finite abelian CM-extension, let $p$ be an odd prime, and put $r=\#S'_f-1$. Let $\operatorname{Ker}(\theta_V)_H$ be the theta-kernel module, let $\psi_{\overline{\Sigma},K}^{\Sigma'}$ be the map defined in the source for an intermediate CM-field $K$ of $H/F$, and let $\Theta_{\overline{\Sigma}}^{\Sigma'}(K/F)$ be the corresponding equivariant $L$-value. Kurihara's determinant conjecture. There is a basis $\{e_{i,H}\}_{1\leq i\leq r}$ of ${}_p\operatorname{Ker}(\theta_V)_H^{-}$ over $\mathbb{Z}_p[G]^{-}$ such that, after defining $e_{i,K}$ by restriction to every intermediate CM-field $K$, for every such $K$ and every subset $\overline{\Sigma}\subseteq\Sigma$ with $\overline{\Sigma}\supset S_\infty(F)\cup S_{\operatorname{ram}}(K/F)$, the determinant of $\psi_{\overline{\Sigma},K}^{\Sigma'}$ with respect to the induced bases satisfies

$$\det(\psi_{\overline{\Sigma},K}^{\Sigma'})=\Theta_{\overline{\Sigma}}^{\Sigma'}(K/F).$$

The source attributes this formulation to Kurihara and uses it to relate determinant and Fitting-ideal formulations of the minus eTNC; no resolution is given in the supplied text.