Stark's conjecture on Stark elements

Let $K/k$ be an abelian extension of number fields with Galois group $G$, let $S$ satisfy (St1)–(St3), and let $w$ be a place of $K$ in $S_K$ with trivial decomposition group in $G$; write $v$ for the place of $k$ below $w$, let $e$ be the number of roots of unity in $K$, and let $U_{K/k}^{\mathrm{ab}}$ and $U^{(v)}$ be the submodules of $\mathcal{O}_{K,S}^{\times}$ defined in the source. Stark element conjecture. There is $\epsilon\in U_{K/k}^{\mathrm{ab}}\cap U^{(v)}$ such that

$$\log\|\epsilon\|_{\sigma w}=-e\,\zeta'_{K/k,S}(0,\sigma^{-1})\quad\text{for all }\sigma\in G,$$

equivalently,

$$L'_{K/k,S}(0,\chi)=-\frac{1}{e}\sum_{\sigma\in G}\overline{\chi}(\sigma)\log\|\epsilon\|_{\sigma w}\quad\text{for all }\chi\in\widehat{G},$$

and

$$\sum_{\chi\in\widehat{G}}L'_{K/k,S}(0,\chi)e_{\overline{\chi}}=-\frac{1}{e}\sum_{\sigma\in G}\log\|\epsilon\|_{\sigma w}\,\sigma.$$

These Stark elements are described as existing only conjecturally in general, although they are known in some cases; the source attributes the formulation to Stark's conjecture as presented by Tate.