The Brumer–Stark conjecture for partial zeta values

Let $F$ be totally real, let $H/F$ be a finite abelian CM extension with Galois group $G$, and let $S=R\cup\{\mathfrak{p}\}$, where $\mathfrak{p}$ splits completely in $H/F$. Let $T$ be an auxiliary finite set of primes satisfying the stated integrality hypotheses, let $U_{\mathfrak{p}}$ be the group of elements of $H^*$ whose absolute values are $1$ at every place not dividing $\mathfrak{p}$, and let $\mathfrak{P}$ be a prime of $H$ above $\mathfrak{p}$. The Brumer–Stark conjecture. There exists an element $u_T\in U_{\mathfrak{p}}$ such that $u_T\equiv 1\pmod{T}$ and, for every $\sigma\in G$,

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

This is the Brumer–Stark conjecture in Gross's formulation, relating valuations of a global unit to integral partial zeta values. It has been proved away from $2$ by Dasgupta–Kakde and over $\mathbb{Z}$ by Dasgupta–Kakde–Silliman–Wang.