Gross's conjecture on the Brumer–Stark reciprocity law

Let $F$, $H$, $S$, $T$, and $\mathfrak{p}$ be as above, and let $L/F$ be a finite abelian CM extension containing $H$ and unramified outside $S_{\mathfrak{p}}=S\cup\{\mathfrak{p}\}$. Write $\mathfrak{g}=\operatorname{Gal}(L/F)$, $\Gamma=\operatorname{Gal}(L/H)$, $G=\operatorname{Gal}(H/F)$, and let $I$ be the relative augmentation ideal, the kernel of $\mathbf{Z}[\mathfrak{g}]\twoheadrightarrow\mathbf{Z}[G]$. For the global reciprocity map $\operatorname{rec}_{\mathfrak{P}}:H_{\mathfrak{P}}^*\to\Gamma$, define

$$\operatorname{rec}_G(u_{\mathfrak{p}})=\sum_{\sigma\in G}(\operatorname{rec}_{\mathfrak{P}}\sigma(u_{\mathfrak{p}})-1)\widetilde{\sigma}^{-1}\in I/I^2,$$

where $\widetilde{\sigma}$ is any lift of $\sigma$ to $\mathfrak{g}$. Gross's conjecture. One has

$$\operatorname{rec}_G(u_{\mathfrak{p}})\equiv\Theta_{S_{\mathfrak{p}},T}^{L/F}\pmod {I^2}$$

in $I/I^2$. The source proves this congruence in $(I/I^2)\otimes\mathbf{Z}_p$ when the rational prime below $\mathfrak{p}$ is odd; the integral statement is therefore the remaining conjectural assertion.