The refined Rubin–Stark reciprocity conjecture

Let $p$ be an odd prime splitting completely in $k$, let $K/k$ and the other notation be as in the paper, and let $G=\operatorname{Gal}(K/k)$. Let $\eta^+_{K/k}$ be the image of the element supplied by Rubin's Conjecture A', let ${\mathcal I}(\eta^+_{K/k})$ be the finite-index ideal defined in the paper, let $\kappa_n^*$ and $\overline{{\mathfrak s}_{K/k}(\theta)}$ be the reduction and reciprocity maps used there, and let ${\mathcal H}_n$ be the corresponding higher Hilbert pairing. The refined Rubin–Stark reciprocity conjecture. For every $x\in{\mathcal I}(\eta^+_{K/k})$ there exists $\widetilde\eta_x\in\bigwedge^d_{{\mathbb Z}G}K^\times$ such that $x\eta^+_{K/k}=1\otimes\widetilde\eta_x$ and

$$\kappa_n^*\bigl(x\overline{{\mathfrak s}_{K/k}(\theta)}\bigr)={\mathcal H}_n(\widetilde\eta_x,\theta)$$

in $({\mathbb Z}/p^{n+1}{\mathbb Z})G$ for all $\theta\in\bigwedge^d_{{\mathbb Z}_pG}U_p^1(K)$. This is the refined reciprocity statement sought after the rational Rubin–Stark element is introduced. The paper formulates it as a conjectural identity; no resolution is supplied.