The local Tamagawa number conjecture for $h^0(K)(j)$

Let $K$ be a number field, let $p$ be a prime, and let $j$ be a positive integer. Write $K_p=K\otimes_{\mathbb{Q}}\mathbb{Q}_p$, let $\mathbb{Z}_p(j)$ denote the $j$-th Tate twist, and let $\vartheta_j^{\rm loc}$ be the local period isomorphism defined in the preceding construction. A $\mathbb{Z}_p$-basis means a generator of the displayed rank-one determinant module.

The local Tamagawa number conjecture. For every positive integer $j$ there exists a unique $\mathbb{Z}_p$-basis

$$z_j \in {\det}_{\mathbb{Z}_p}^{-1}(\operatorname{R}\Gamma(K_p,\mathbb{Z}_p(j))) \otimes_{\mathbb{Z}_p} {\det}^{-1}_{\mathbb{Z}_p} \left( \bigoplus_{\sigma: K \hookrightarrow \mathbb{C}} \mathbb{Z}_p(j)\right)$$

with

$$\vartheta_j^{\rm loc}(z_j)= \frac{\zeta_{K, \{p\}}^\ast(1-j)}{\zeta_{K,\{p\}}^\ast(j)}.$$

This is the local Tamagawa number conjecture for the pair $(h^0(K)(j),p)$; the supplied text gives no resolution beyond its formulation.