Gross-Stark conjecture

Assume, in addition to the Stark setup, that $F$ is totally real, $K$ is a CM-field, $v$ is a finite place above a rational prime $p$, and $S$ contains all places of $F$ above $p$. Let $\zeta_{S,p}(s,\sigma)$ be the $p$-adic interpolation function of $\zeta_S(s,\sigma)$, and let $\epsilon'\in K$ be a $v$-unit satisfying $\log|\epsilon'^\sigma|_w=-W\zeta'_S(0,\sigma)$. Gross-Stark conjecture.

$$\log_p N_{K_w/\mathbb Q_p}(\epsilon'^\sigma)=-W\zeta'_{S,p}(0,\sigma).$$

This is the $p$-adic analogue of the Stark unit formula; the supplied text does not state whether it is open or resolved.