Oda–Yoshida's archimedean isogeny conjecture for Darmon points

Let $F$ have mixed signature, let $v$ be the distinguished real place, and let $L\subset\mathbb C$ be the lattice obtained from the integration pairing and the homology connecting map. Let $E_v/\mathbb C$ be the base change of $E$ at $v$.

Oda–Yoshida isogeny conjecture. There exists an isogeny

$$\beta\colon \mathbb C/L\longrightarrow E_v(\mathbb C).$$

This generalizes archimedean conjectures of Oda and Yoshida to mixed-signature fields. The paper notes that it amounts to the existence of a nonzero complex number carrying the relevant period lattice into $L$; it is not proved in general.