Global Selmer class conjecture for twisted Stark–Heegner cycles

Let $\chi$ be a character of $\operatorname{Gal}(H_{\mathcal{C}}/K)$, let $H_{\chi}$ be the abelian subextension cut out by $\chi$, and let $\mathcal{D}_{\chi}$ be the corresponding $\chi$-twisted Stark–Heegner cycle. Let $\mathcal{S}_{\chi}$ denote a class in the $\chi$-isotypic component of the semistable Selmer group.

Twisted Stark–Heegner cycle conjecture. There exists a global Selmer class

$$\mathcal{S}_{\chi} \in \operatorname{Sel}_{\mathrm{st}}(H_{\chi}, V_{p}(\mathcal{F})(k_{0}/2+1))^{\chi}$$

such that

$$\operatorname{exp}_{\mathrm{BK}} \circ \varphi\left(\Phi^{\mathrm{AJ}}(\mathcal{D}_{\chi})\right)=\operatorname{res}_{\mathfrak{p}}(\mathcal{S}_{\chi}),$$

where the superscript $\chi$ denotes the $\chi$-isotypic component.

This is the precise character-by-character formulation of the expected rationality of Stark–Heegner cycles. The supplied text gives no evidence that it has been resolved.