Stark's conjecture over the integers for higher-rank Rubin–Stark elements

Let $K/k$ be an abelian extension with Galois group $\Gamma$, and let $S$, $T$, and $S'$ satisfy the following conditions: $S$ contains all archimedean places and all places ramified in $K/k$; $T$ is disjoint from $S$ and $U_{K,S,T}$ contains no roots of unity; and $S'\subsetneq S$ consists only of places that split completely in $K$. Put $r=|S'|$, and let $W^*_{K,S'}$ and $\rho_{K,r}$ be as defined in the source. The regulator map is $\mathcal{R}^\infty:\bigl(\bigwedge^r U_{K,S,T}\bigr)\otimes_{\Gamma}\bigwedge^rW^*_{K,S'}\to\mathbb{R}[\Gamma]$.

Stark's conjecture over $\mathbb{Z}$. There is a unique element

$$\epsilon_K=\epsilon_{K,S,T,S'}\in\bigl(\bigwedge^r U_{K,S,T}\bigr)[\rho_{K,r}]\otimes_{\Gamma}\bigwedge^rW^*_{K,S'}$$

such that

$$\mathcal{R}^\infty(\epsilon_K)=\lim_{s\to0}s^{-r}\theta_{K/k}(s).$$

This refines Stark's conjecture by predicting an integral exterior-power element whose regulator gives the leading term of the equivariant $L$-function. It is known when $r=0$ and when $K/k$ is quadratic, but is open in general.