Stark's rank one abelian conjecture for quadratic fields

Let $F$ be a quadratic extension of $\mathbb{Q}$ and let $K$ be a nontrivial finite abelian extension of $F$. If $F$ is real quadratic, assume that one infinite place of $F$ stays real in $K$ and the other becomes complex. Let $S$ be a finite set of places of $F$ containing the infinite places and the places ramified in $K$, with $|S|\geq 2$, and let $S_K$ be the places of $K$ above $S$. Choose an infinite place $v$ of $K$ such that $v(K)\subset\mathbb{R}$ when $F$ is real quadratic, and also write $v$ for its restriction to $F$. Let $U_{v,S}\subset K^\times$ be the subgroup defined by

$$U_{v,S}=\begin{cases}\{u\in K^\times: |u|_{w'}=1,\ \forall w'\text{ such that }w'\rvert_F\not=v\rvert_F\},& |S|\geq 3,\\ \{u\in K^\times: |u|_{w'}=|u|_{w”},\ \forall w',w”\mid v'\text{ and }|u|_w=1,\ \forall w\notin S_K\},& S=\{v,v'\}.\end{cases}$$

Let $e$ be the number of roots of unity in $K$, and let $L_S(\chi,s)$ be the complex $L$-function associated to a character $\chi$ of $\operatorname{Gal}(K/F)$ with the Euler factors at the primes in $S$ removed. Stark's conjecture. There exists $u\in U_{v,S}$ such that for all characters $\chi$ of $\operatorname{Gal}(K/F)$,

$$L'_S(\chi,0)=-\frac{1}{e}\sum_{\sigma\in\operatorname{Gal}(K/F)}\chi(\sigma)\log|\sigma(u)|_v.$$

Stark proved this conjecture when $F$ is imaginary quadratic, whereas it remains open when $F$ is real quadratic. The additional conclusion that $K(u^{1/e})$ is abelian over $F$ is not included here.