Gross–Zagier non-vanishing conjecture for Darmon cycles

Let $f$ be a $p$-new eigenform, let $\mathfrak{p}$ be a prime above $p$, let $V_{\mathfrak{p}}(f)$ be the corresponding $p$-adic representation, let ${\Greekmath 011F}$ be a character of the relevant ring class group, and let $\underline{s}_{f,\mathfrak{p}}^{\Greekmath 011F}$ and $s_{f,\mathfrak{p}}^{\Greekmath 011F}$ be respectively the conjectural global and Darmon cohomology classes. Write $L(f/K,{\Greekmath 011F},s)$ for the Rankin–Selberg $L$-function. Gross–Zagier non-vanishing conjecture.

$$\underline{s}_{f,\mathfrak{p}}^{\Greekmath 011F}\neq 0\Longleftrightarrow L'(f/K,{\Greekmath 011F},k/2)\neq 0,$$

and in particular

$$s_{f,\mathfrak{p}}^{\Greekmath 011F}\neq 0\Longrightarrow L'(f/K,{\Greekmath 011F},k/2)\neq 0.$$

This is proposed as a Gross–Zagier-type statement for Darmon cycles. The paper notes that even the corresponding conjectures for classical Heegner cycles remain completely open, so no resolution is supplied here.