The non-archimedean globality and reciprocity conjecture for Darmon points

Let $K_{\mathfrak p}$ be the relevant completion, let $E/K_{\mathfrak p}$ be the elliptic curve, and let $H_{\mathcal O}^+$ be the narrow ring class field embedded in $K_{\mathfrak p}$. Let $P_\psi$ be the local point obtained from the integration pairing and an isogeny $\beta\colon K_{\mathfrak p}^\times/L\to E(K_{\mathfrak p})$. Let $\operatorname{rec}\colon\operatorname{Pic}^+({\mathcal O})\to\operatorname{Gal}(H_{\mathcal O}^+/K)$ be the reciprocity map.

Non-archimedean Darmon point conjecture. The local point $P_\psi$ is rational over $H_{\mathcal O}^+$, and, for every $\alpha\in\operatorname{Pic}^+({\mathcal O})$,

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

This is the non-archimedean mixed-signature generalization of conjectures of Greenberg, Darmon, and Trifković. The asserted globality and reciprocity law are open in the general setting.