The globality and reciprocity conjecture for mixed-signature Darmon points

Let $F$ be a number field of mixed signature, let $E/F$ be the elliptic curve in the construction, let ${\mathcal O}$ be an order in a quadratic extension $K/F$, and let $H_{\mathcal O}^+$ be its narrow ring class field. Let $\operatorname{Pic}^+({\mathcal O})$ act on the set of optimal embeddings, let $\operatorname{rec}\colon \operatorname{Pic}^+({\mathcal O})\simeq\operatorname{Gal}(H_{\mathcal O}^+/K)$ be reciprocity, and let $P_\psi$ be the point obtained from an isogeny $\beta$ as above.

Darmon point globality conjecture. The isogeny $\beta$ can be chosen so that $P_\psi$ is the image of a global point in $E(H_{\mathcal O}^+)$, and, for every $\alpha\in\operatorname{Pic}^+({\mathcal O})$,

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

This extends conjectures of Darmon, Gärtner, Greenberg, and Trifković to mixed-signature base fields. The assertion that the locally constructed points are global, together with the reciprocity law, remains conjectural in this generality.