Rank-one abelian Stark conjecture

Let $K/F$ be an abelian extension of number fields, let $S$ be a finite set of places containing all infinite places of $F$ and all places ramified in $K/F$, and suppose that a distinguished place $v\in S$ splits completely in $K/F$, with a fixed place $w$ of $K$ above $v$, and that $|S|\geq 2$. Let $\zeta_S(s,\sigma)$ be the associated partial zeta function, and let $W_K$ denote the number of roots of unity in $K^\times$. The rank-one abelian Stark conjecture. There exists an element $\epsilon\in K^\times$ such that: if $|S|>2$, then $\epsilon$ is a $v$-unit; if $S=\{v,v'\}$, then $\epsilon$ is an $S$-unit and $|\epsilon|_{w'}$ is constant for every place $w'$ of $K$ above $v'$; moreover,

$$\log |\epsilon^\sigma|_w=-W_K\zeta'_S(0,\sigma)\qquad(\sigma\in\operatorname{Gal}(K/F)),$$

and $K(\epsilon^{1/W_K})/F$ is abelian. This is the rank-one form of Stark's conjecture, relating special derivatives of partial zeta functions to algebraic units; its status is not specified in the supplied text.