The Brumer–Stark conjecture for Brumer–Stark units

Let $F$ be a totally real field, let $H/F$ be a finite abelian extension, and let $G=\operatorname{Gal}(H/F)$. Let $R$ contain the archimedean places of $F$ and the places ramified in $H$, let $\mathfrak{p}\notin R$ split completely in $H$, put $S=R\cup\{\mathfrak{p}\}$, and let $T$ be an auxiliary finite set disjoint from $R\cup\{\mathfrak{p}\}$. For a prime $\mathfrak{P}$ of $H$ above $\mathfrak{p}$, define

$$\mathcal{U}_{\mathfrak{p}}=\{u\in H^*: |u|_v=1\text{ if }v\text{ does not divide }\mathfrak{p}\}.$$

Brumer–Stark conjecture. There exists $u_T\in\mathcal{U}_{\mathfrak{p}}$ such that $u_T\equiv1\pmod T$ and, for every $\sigma\in G$,

$$\operatorname{ord}_{\mathfrak{P}}(u_T^\sigma)=\zeta_{R,T}(H/F,\sigma,0).$$

Here the absolute-value condition ranges over all finite and archimedean places of $H$, and $\zeta_{R,T}(H/F,\sigma,0)$ is the relevant partial zeta value. The conjecture has recently been proved away from $2$ in joint work of the first author and Kakde; thus its remaining status is confined to the $2$-part.