Narrow ring class field reciprocity conjecture for Darmon points

Let $K/F$ be a quadratic extension, let $\mathcal{O}$ be an order in $K$, let $K_{\mathcal{O}}^+$ be the associated narrow ring class field, and let $P_{q,b}$ be the Darmon point associated with the construction. Let $\operatorname{Pic}(\mathcal{O})^+$ be the narrow Picard group acting on the points, and let

$$\operatorname{rec}:\operatorname{Pic}(\mathcal{O})^+\longrightarrow\operatorname{Gal}(K_{\mathcal{O}}^+/K)$$

be the reciprocity isomorphism. Narrow reciprocity conjecture. The point $P_{q,b}$ is defined over $K_{\mathcal{O}}^+$, and for every $\alpha\in\operatorname{Pic}(\mathcal{O})^+$,

$$\operatorname{rec}(\alpha)(P_{q,b})=\alpha*P_{q,b}.$$

This gives the predicted field of definition and Galois action for the Darmon points constructed in the paper.