Gross's explicit reciprocity conjecture for Brumer–Stark units

Let $H/F$ be as in the Brumer–Stark setting, let $L/F$ be a finite abelian CM extension containing $H$, and write $\mathfrak{g}=\operatorname{Gal}(L/F)$ and $\Gamma=\operatorname{Gal}(L/H)$. Let $I$ be the relative augmentation ideal for $\mathfrak{g}\twoheadrightarrow G=\operatorname{Gal}(H/F)$, and let $u_{\mathfrak p}$ be the Brumer–Stark unit attached to a prime $\mathfrak p$ splitting completely in $H$. 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 $\sigma$. Gross's conjecture. One should have

$$\operatorname{rec}_G(u_{\mathfrak p})\equiv\Theta_{S',T}^{L/F}\pmod{I^2},$$

viewed as an equality in $I/I^2$. This conjecture gives an explicit reciprocity-law interpretation of the first-order augmentation derivative of a Stickelberger element in terms of Brumer–Stark units. The source gives no resolution status.