Non-triviality conjecture for Darmon cohomology classes

Let f\in S_k(\Gamma_0(pN^+))^{p\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-new}} be a $p$-new eigenform, let $L_{f,\mathfrak{p}}$ be the completion of its coefficient field at a prime $\mathfrak{p}$ above $p$, let $V_{\mathfrak{p}}(f)$ be the corresponding summand of the $p$-adic representation, and let $H_{\Greekmath 011F}/K$ be the abelian extension cut out by a character ${\Greekmath 011F}$. Let $s_{f,\mathfrak{p}}^{\Greekmath 011F}$ be the Darmon class and $\underline{s}_{f,\mathfrak{p}}^{\Greekmath 011F}$ its conjectural global lift. Non-trivial cases conjecture. If $s_{f,\mathfrak{p}}^{\Greekmath 011F}\neq 0$, then

$$\operatorname{MW}(H_{\Greekmath 011F},V_{\mathfrak{p}}(f))^{\Greekmath 011F}=L_{f,\mathfrak{p}}\underline{s}_{f,\mathfrak{p}}^{\Greekmath 011F}.$$

This predicts that, whenever the local Darmon class is nonzero, the corresponding global ${\Greekmath 011F}$-isotypical Mordell–Weil space is one-dimensional over $L_{f,\mathfrak{p}}$ and generated by the global class. The text gives no resolution.