Brumer–Stark conjecture for Abelian extensions

Let $K/k$ be an Abelian extension of number fields, let $G=\operatorname{Gal}(K/k)$, and let $w_K$ be the number of roots of unity in $K$. Set

$$\theta=\theta_{K/k,S}(0).$$

An element of $K^{\times}$ is an anti-unit if all of its complex absolute values are $1$, and it is $w_K$-Abelian for $K/k$ if adjoining a $w_K$-th root produces an Abelian extension of $k$. If $\mathfrak{a}$ is an integral ideal of $K$, then:

Brumer–Stark conjecture. There exists $\varepsilon_{\mathfrak{a},K/k,S}\in K^{\times}$ such that

$$\mathfrak{a}^{w_K\theta}=\left(\varepsilon_{\mathfrak{a},K/k,S}\right),$$

$\varepsilon_{\mathfrak{a},K/k,S}$ is an anti-unit, and $\varepsilon_{\mathfrak{a},K/k,S}$ is $w_K$-Abelian.

This is the $p$-local Brumer–Stark conjecture used as a hypothesis for the paper's results. The supplied text does not state a resolution.