Greenberg's rationality conjecture for the point $P_\psi$

Let $E$ be an elliptic curve over ${\mathbb Q}$, let $K$ be the real quadratic field in the preceding construction, and let $H_K^+$ denote its narrow Hilbert class field. With $J_\psi$ the integration pairing defining a class in $K_p^\times/L$, let $\beta:K_p^\times/L\to K_p^\times/\langle q_E\rangle$ be the isogeny to the Tate quotient and let $\Phi_{\operatorname{Tate}}$ be Tate's uniformization map. Define

P_\psi=(\Phi_{\operatorname{Tate}}\begin{picture}(6,5)(-3,-2)\put(0,1){\circle{2}} \end{picture}\beta)(J_\psi)\in E(K_p).

Greenberg's rationality conjecture. The local point $P_\psi$ is a global point, and more precisely

$$P_\psi\in E(H_K^+).$$

This is the explicit narrow-Hilbert-class-field formulation of the quaternionic Darmon-point rationality prediction. The supplied excerpt states it as a conjecture but provides no evidence of a general proof or disproof.