Tate–Gross conjecture on Gross–Stark units

Let $K/F$ be the abelian extension with Galois group $G$, let $S$ and $T$ be the sets of places used to define the elements $u_{S,T,\sigma}$, let ${\mathscr E}_{\mathfrak p}$ be the subgroup of $K^*$ consisting of elements having absolute value $1$ at every place away from ${\mathfrak p}$, and let $H$ be the field appearing in the definition of the congruence condition. The elements $u_{S,T,\sigma}\in F_{\mathfrak p}^*$ are defined by

$$u_{K/F,S,T}=\sum_{\sigma\in G}u_{S,T,\sigma}\otimes[\sigma^{-1}].$$

Tate–Gross conjecture. For every $\sigma\in G$, $u_{S,T,\sigma}\in{\mathscr E}_{\mathfrak p}$; for every $\sigma\in G$ and every ${\mathfrak q}\in T$, $u_{S,T,\sigma}\equiv1\pmod{{\mathfrak q}{\mathcal O}_H}$; and for all $\sigma,\tau\in G$, $\tau(u_{S,T,\sigma})=u_{S,T,\tau\sigma}$. This conjecture strengthens the Brumer–Stark conjecture in Tate's formulation and a conjecture of Gross. The supplied text gives no resolution status.