Gross's integral Gross–Stark conjecture

Let $L/H/F$ be a tower in which $F$ is totally real and $H$ and $L$ are finite abelian CM extensions of $F$, with $L$ containing $H$. Write $\mathcal{G}=\operatorname{Gal}(L/F)$, $G=\operatorname{Gal}(H/F)$, and let $S=R\cup\{\mathfrak{p}\}$, where $\mathfrak{p}$ splits completely in $H/F$. Define

$$\Theta_{S,T}^{L/F}=\sum_{\sigma\in G}\zeta_{S,T}(L/F,\sigma,0)\otimes\sigma^{-1}\in\mathbb{C}[\mathcal{G}],$$

let $I$ be the relative augmentation ideal, and let $\operatorname{rec}_{\mathfrak{p}}:F_{\mathfrak{p}}^*\to\mathcal{G}$ be the local reciprocity map. For $u_{\mathfrak{p}}=\sum_{\sigma\in G}u_{\mathfrak{p}}(\sigma)\otimes\sigma^{-1}$, define

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

where $\widetilde{\sigma}\in\mathcal{G}$ is any lift of $\sigma$. Gross's integral Gross–Stark conjecture. One has

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

in $I/I^2$. This conjecture is an integral refinement of the Gross–Stark conjecture, relating reciprocity images of Brumer–Stark units to Stickelberger elements. The source gives no resolution status for this conjecture.