The archimedean globality and reciprocity conjecture for Darmon points

Let $F$ have mixed signature, let $v$ be the distinguished real place, let ${\mathcal O}$ be an order in a quadratic extension $K/F$, and let $H_{\mathcal O}^+$ be its narrow ring class field embedded in $\mathbb C$. Let $P_\psi$ be the point obtained from an isogeny $\beta\colon\mathbb C/L\to E_v(\mathbb C)$, and let $\operatorname{rec}\colon\operatorname{Pic}^+({\mathcal O})\simeq\operatorname{Gal}(H_{\mathcal O}^+/K)$ be reciprocity.

Archimedean Darmon point conjecture. The isogeny $\beta$ can be chosen so that

$$P_\psi\in E_v(H_{\mathcal O}^+)$$

and, for every $\alpha\in\operatorname{Pic}^+({\mathcal O})$,

$$P_{\alpha\cdot\psi}=\operatorname{rec}(\alpha)(P_\psi).$$

This is described as a mixed-signature generalization of conjectures of Gärtner and Darmon. The globality and reciprocity properties remain conjectural.