Rubin's Conjecture A' for higher Stark regulators

Let $K^+/k$ be the totally real subextension under consideration, let $G^+=\operatorname{Gal}(K^+/k)$, let $S_0=S_{\rm ram}(K/k)$, and let $U_{S_0}(K^+)$ be the group of global $S_0$-units of $K^+$. Define the archimedean regulator

$$R_{K^+/k}:{\mathbb Q}\otimes\bigwedge^d_{{\mathbb Z}G^+}U_{S_0}(K^+)\longrightarrow {\mathbb R}G^+$$

by sending $u_1\wedge\cdots\wedge u_d$ to the determinant of the logarithmic maps defined in the paper. Let $\Theta_{K^+/k,S_0}^{(d)}(0)$ be the leading $d$-th derivative term at $s=0$, let $e_\chi$ be the character idempotent, and let $X(S_0,d,G^+)$ be the characters for which $\chi(\Theta_{K^+/k,S_0}^{(d)}(0))\neq0$. Rubin's Conjecture A'. There exists an element $\eta_{K^+/k,S_0}$ of

$${\mathbb Q}\otimes\bigwedge^d_{{\mathbb Z}G^+}U_{S_0}(K^+)$$

such that

$$\Theta_{K^+/k,S_0}^{(d)}(0)=R_{K^+/k}(\eta_{K^+/k,S_0})$$

and

$$e_\chi\eta_{K^+/k,S_0}=0$$

in ${\mathbb C}\otimes\bigwedge^d_{{\mathbb Z}G^+}U_{S_0}(K^+)$ for every character $\chi$ of $G^+$ not in $X(S_0,d,G^+)$. This is the rational form of Rubin's conjecture and is equivalent here to the relevant instances of Stark's conjecture. The source subsequently assumes it and notes that it holds in the cyclotomic example, but gives no general proof.