The $p$-adic Stark-unit conjecture for Gross's $p$-adic regulator

Let $F$ be a totally real field, let $K$ be a CM-field abelian over $F$, and let $S$ contain all places above $p$. Assume that $\mathfrak p_F$ splits completely in $K/F$, put $\mathfrak p_K$ for the place induced by the fixed embedding into $\mathbb C_p$, and let $u'\in K$ be a $\mathfrak p_F$-unit satisfying

$$\log \|{u'}^\tau\|_{\mathfrak p_K}=-W\zeta'_S(0,\tau)\qquad(\tau\in\operatorname{Gal}(K/F)).$$

Gross's $p$-adic conjecture. One has

$$\log_p N_{K_{\mathfrak p_K}/\mathbb Q_p}({u'}^\tau)=-W\zeta'_{p,S}(0,\tau)\qquad(\tau\in\operatorname{Gal}(K/F)).$$

This is the conjectural $p$-adic property of the Stark unit in the CM-field setting and refines the rank-one abelian Stark conjecture. The source gives no resolution status.