A part of Stark's conjecture for totally real abelian extensions

Let $F$ be totally real, let $K$ be a finite abelian extension of $F$ with $G:=\operatorname{Gal}(K/F)$, and assume $K\neq\mathbb Q$ and that $K$ admits an embedding into $\mathbb R$, which we use to regard $K$ as a subfield of $\mathbb R$. Let $S_0$ be the union of the infinite places and the ramified places, and write

$$\zeta(s,\sigma):=\zeta_{S_0}(s,\sigma).$$

A part of Stark's conjecture. For every $\sigma\in G$,

$$u_F(\sigma):=\exp(-2\zeta'(0,\sigma))\in\overline{\mathbb Q}^{\times},$$

and for every $\sigma\in G$ and $\tau\in G_F:=\operatorname{Gal}(\overline{\mathbb Q}/F)$,

$$\tau(u_F(\sigma))=u_F(\tau\sigma).$$

These are the algebraicity and reciprocity components obtained from the rank $1$ abelian Stark conjecture in the real-place setting. The paper focuses on proving these conditions in a partial form via reciprocity laws for period ring-valued beta functions.