Yuan–Zhang–Zhang-type nonvanishing conjecture for Darmon's points

Let $\chi$ be a character of $\operatorname{Gal}(K_b^+/K)$, let $K_{\chi}=(K^{\mathrm{ab}})^{\operatorname{Ker}(\chi)}$, and let $\ell_{\chi}$ be the associated linear functional. Assume

$$\forall v\neq\tau_1\quad \varepsilon(\pi_v\times\chi_v,\tfrac12)\eta_{K,v}(-1)=\operatorname{inv}_v(B).$$

Yuan–Zhang–Zhang-type conjecture. There exists $b\in\widehat{B}^{\times}$ with $k(b,\omega_{\varphi})\neq0$ such that

$$\ell_{\chi}\neq0\Longleftrightarrow\exists b\in B_{\mathbf A}^{\times}\ \mathrm{such\ that}\ K_{\chi}\subset K_b^+\ \mathrm{and}\ e_{\overline{\chi}}(P_b^{\chi_{\infty}})\in\mathbf Z[\chi]\otimes E(K_b^+)\ \mathrm{is\ not\ torsion}$$ $$\Longleftrightarrow\exists\sigma:\mathbf Q(\chi)\hookrightarrow\mathbf C\quad L'(\pi\times{}^{\sigma}\!\chi,\tfrac12)\neq0$$ $$\Longleftrightarrow\forall\sigma:\mathbf Q(\chi)\hookrightarrow\mathbf C\quad L'(\pi\times{}^{\sigma}\!\chi,\tfrac12)\neq0.$$

This relates nonvanishing of Darmon-point projections to derivatives of Rankin–Selberg $L$-functions, analogously to Gross–Zagier formulas; the supplied source gives no resolution.