Darmon's conjecture on ATR points and analytic rank

Let $F$ be totally real, let $K/F$ be a quadratic ATR extension, let $H_{\mathfrak c}$ be the ring class field of an order in $K$, and let $P_\varphi$ be the ATR point associated with an optimal embedding. Let

$$P_K=\sum_{\varphi\in \mathcal E_{\mathfrak c}/\widetilde\Gamma}P_\varphi.$$

Here $\mathcal E_{\mathfrak c}/\widetilde\Gamma$ is the set of equivalence classes of the relevant optimal embeddings.

Darmon's conjecture. The point $P_\varphi$ belongs to $E_0(H_{\mathfrak c})$, the point $P_K$ belongs to $E_0(K)$, and $P_K$ is non-torsion if and only if

$$\operatorname{ord}_{s=1}L(E/K,s)=1.$$

This is the ATR analogue of the Heegner-point relationship between algebraic points and analytic rank. The source states it conditionally on Oda's conjecture and gives no resolution status.