Rubin–Stark conjecture on the integrality of Rubin–Stark elements

Let $K/k$ be the abelian extension of global fields with Galois group $G$, and let $S$, $T$, and $V$ be the auxiliary sets of places used to define the Rubin–Stark element $\eta_{K/k,S,T}^r$. Let $O_{K,S,T}^{\times}$ be the corresponding group of $(S,T)$-units, and define Rubin's lattice by

$$\bigcap_{\mathbb{Z}[G]}^r O_{K,S,T}^{\times}:=\left(\bigwedge_{\mathbb{Z}[G]}^r (O_{K,S,T}^{\times})^*\right)^*,$$

where $(O_{K,S,T}^{\times})^*=\operatorname{Hom}_{\mathbb{Z}[G]}(O_{K,S,T}^{\times},\mathbb{Z}[G])$.

Rubin–Stark conjecture. The Rubin–Stark element satisfies

$$\eta_{K/k,S,T}^r\in\bigcap_{\mathbb{Z}[G]}^r O_{K,S,T}^{\times}.$$

This is the integrality assertion that the analytically defined Rubin–Stark element lies in the associated exterior power bidual, or Rubin's lattice, rather than only in its rational extension. The supplied context does not indicate whether this assertion is known in the stated generality.