The local equivariant Tamagawa number conjecture for Tate motives

Let $p$ be an odd prime, let $K/k$ be a finite abelian extension of number fields with Galois group $G$, and let $S$ contain $S_p\cup S_\infty\cup S_{\operatorname{ram}}(K/k)$, with $S_f=S\setminus S_\infty$. For $j\in\mathbb{Z}_{\geq 1}$, let $\Xi_{K/k,S}^{\operatorname{loc}}(j)$ and $\vartheta_{K/k,S}^{\operatorname{loc},j}$ be the determinant module and its specified isomorphism to $\mathbb{C}_p[G]$. For each finite prime $v$ of $k$, let $\delta_v(K/k)$ be the element defined from the Euler factor, and let $\delta_{K/k,{\bf{1}}}$ be the element whose value under the trivial character is $-1$ and under every nontrivial character is $1$. Also write $\Theta_{K/k,S}^\ast(j)$ for the leading-term element and $r_k$ for the number of real places of $k$. The local equivariant Tamagawa number conjecture. For any $j\in\mathbb{Z}_{\geq 1}$, there is a unique $\mathbb{Z}_p[G]$-basis $z_{K/k,S}^{\operatorname{loc},j}\in\Xi_{K/k,S}^{\operatorname{loc}}(j)$ such that

$$\vartheta_{K/k,S}^{\operatorname{loc},j}(z_{K/k,S}^{\operatorname{loc},j})=(-1)^{r_k(j-1)}\times\begin{cases}\displaystyle\frac{\Theta_{K/k,S}^\ast(1-j)^\#}{\Theta_{K/k,S}^\ast(j)}&\text{if }j\geq2,\\[6pt]\displaystyle\delta_{K/k,{\bf{1}}}\cdot\left(\prod_{v\in S_f}\delta_v(K/k)^\#\right)\frac{\Theta_{K/k}^\ast(0)^\#}{\Theta_{K/k,S}^\ast(1)}&\text{if }j=1.\end{cases}$$

This is the equivariant refinement of the local Tamagawa number conjecture for the Tate motives $\mathbb{Z}_p(j)$, incorporating the delicate sign and Euler-factor corrections. The paper proves this assertion under the stated unramified-at-$p$ hypotheses, so the conjecture is treated as solved here.