Rational Stark conjecture for a character at s = 0

Keep the setting and notation above for $K/F$, $S$, and $v$. Let $\chi$ be a character of $\operatorname{Gal}(K/F)$ such that $\operatorname{ord}_{s=0}(L_S(\chi,s))=1$, and let $k$ be the field obtained by adjoining the values of $\chi$ to $\mathbb{Q}$. Extend $\log|\cdot|_v$ from $U_{v,S}$ to $k\otimes_{\mathbb{Z}}U_{v,S}$ by $k$-linearity, and define

$$(k\otimes_{\mathbb{Z}}U_{v,S})^{\chi^{-1}}=\{u\in k\otimes_{\mathbb{Z}}U_{v,S}:\sigma(u)=\chi^{-1}(\sigma)u,\ \forall\sigma\in\operatorname{Gal}(K/F)\}.$$

Rational Stark conjecture. There exists an element $u_\chi\in(k\otimes_{\mathbb{Z}}U_{v,S})^{\chi^{-1}}$ such that

$$L'_S(\chi,0)=\log|u_\chi|_v.$$

This is the characterwise rational formulation of the rank one abelian Stark conjecture, obtained after restricting to a character whose $L$-function has a simple zero at $s=0$. The source gives no resolution beyond presenting it as a conjecture.