The isogeny conjecture for mixed-signature Darmon points

Let $F$ be a number field of mixed signature, let $E/F$ be the elliptic curve in the construction, let $v$ be the distinguished place, and let $K_v$ be the corresponding completion. Let $L\subset K_v^\times$ be the subgroup obtained by integrating the cohomology class $\Phi_E$ over the connecting homomorphism images. For an optimal embedding $\psi$, let $J_\psi\in K_v^\times/L$ be the resulting integration class.

Isogeny conjecture. There exists an isogeny

$$\beta\colon K_v^\times/L\longrightarrow E(K_v).$$

This conjecture is the local analytic ingredient needed to define the associated Darmon point $P_\psi=\beta(J_\psi)$. It extends conjectures of Oda and Yoshida at archimedean places and results of Darmon, Dasgupta--Greenberg, and Longo--Rotger--Vigni in the stated non-archimedean special cases.