Rationality conjecture for Darmon cohomology classes

Let $K$ be the real quadratic field, let $H_{\mathcal{O}}/K$ be the narrow ring class field attached to an order $\mathcal{O}$, let $V_p$ be the relevant $p$-adic representation, and let $\operatorname{res}_p$ denote the restriction map from global Mordell–Weil cohomology to local semistable cohomology. For an optimal embedding $\Psi\in\operatorname{Emb}(\mathcal{O},\mathcal{R})$, an ideal class $\mathfrak{a}\in\operatorname{Pic}(\mathcal{O})$, and a character ${\Greekmath 011F}:\operatorname{Gal}(H_{\mathcal{O}}/K)\to\mathbb{C}^{\times}$, use $s_\Psi$, $s_{\mathfrak{a}\star\Psi}$, and $s_{\Greekmath 011F}$ for the associated Darmon cohomology classes. Rationality conjecture.

1. For every optimal embedding $\Psi\in\operatorname{Emb}(\mathcal{O},\mathcal{R})$, there is a global class $\underline{s}_\Psi\in\operatorname{MW}_*(H_{\mathcal{O}},V_p)$ such that

$$s_\Psi=\operatorname{res}_p(\underline{s}_\Psi).$$

2. For every $\Psi\in\mathpalette\overrightarrow@{\operatorname{Emb}}_+(\mathcal{O},\mathcal{R})$ and every $\mathfrak{a}\in\operatorname{Pic}(\mathcal{O})$, if ${\Greekmath 011B}=\operatorname{rec}(\mathfrak{a})^{-1}\in\operatorname{Gal}(H_{\mathcal{O}}/K)$, then

$$\operatorname{res}_p({}^{{\Greekmath 011B}}\underline{s}_\Psi)=s_{\mathfrak{a}\star\Psi}.$$

3. For every character ${\Greekmath 011F}:\operatorname{Gal}(H_{\mathcal{O}}/K)\to\mathbb{C}^{\times}$, there is a class $\underline{s}_{\Greekmath 011F}\in\operatorname{MW}^{{\Greekmath 011F}}(H_{\Greekmath 011F},V_p)$ such that

$$s_{\Greekmath 011F}=\operatorname{res}_p(\underline{s}_{\Greekmath 011F}),$$

where $H_{\Greekmath 011F}/K$ is the abelian subextension cut out by ${\Greekmath 011F}$ and $\operatorname{MW}^{{\Greekmath 011F}}$ denotes its ${\Greekmath 011F}$-isotypical subspace.

The conjecture predicts that locally defined Darmon classes arise by restriction from global cohomology classes and satisfy the expected reciprocity law. The text explicitly notes that part (3) follows from parts (1) and (2); no resolution is supplied.