The rationality and reciprocity conjecture for Darmon points

Let $\mathcal{O}$ be the quadratic order used to define the Darmon points, let $H$ be its narrow ring class field over the real quadratic field $K$, and let $J^{(\ell)}_\pm$ be the abelian varieties in the sign-eigenspace conjecture. For an embedding $\psi\in\operatorname{Emb}(\mathcal{O},R)$, let $P_\psi^\pm$ denote the corresponding Darmon point, and let

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

be the reciprocity isomorphism. The Darmon-point rationality conjecture. For every $\psi\in\operatorname{Emb}(\mathcal{O},R)$,

$$P_\psi^\pm\in J^{(\ell)}_\pm(H),$$

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

$$P_{\psi^{\mathfrak{a}}}^\pm=\operatorname{rec}(\mathfrak{a})^{-1}\bigl(P_\psi^\pm\bigr).$$

This conjecture predicts that Darmon points are algebraic over the narrow ring class field and satisfy the expected class-field-theoretic reciprocity law; the source gives no resolution status.