Darmon's conjecture on the rationality of Darmon points

Let $E$ be an elliptic curve over $\mathbb Q$ and let $K$ be a real quadratic field satisfying the hypotheses in the construction. Let $q$ be the $p$-adic period of $E$, let $\Phi_{\operatorname{Tate}}\colon K_p^\times/q^{\mathbb Z}\rightarrow E(K_p)$ be Tate's uniformization map, and for $\tau\in\mathcal{H}_p$ let $J_\tau$ be the associated multiplicative integral. Define

$$P_\tau=\Phi_{\operatorname{Tate}}(J_\tau).$$

Darmon's conjecture. The local point $P_\tau=\Phi_{\operatorname{Tate}}(J_\tau)$ belongs to $E(H_\tau)$, where $H_\tau$ is the ring class field associated with the order determined by $\tau$. This is the precise ring-class-field rationality assertion underlying the broader expectation that Darmon points behave like Heegner points. The source gives no resolution, while the paper provides additional computational evidence supporting the conjecture.