Gross–Zagier non-vanishing criterion for Stark–Heegner cycles

Let $\chi:\operatorname{Gal}(H_{\mathcal C}/K)\to\mathbb C^\times$ be a character, let $\mathfrak s_\chi$ be the associated Stark–Heegner cycle class, and let $L(f/K,\chi,s)$ be the corresponding twisted base-change $L$-function. Gross–Zagier non-vanishing conjecture. If

$$\mathfrak s_\chi\neq 0,$$

then

$$L'(f/K,\chi,\tfrac{k+2}{2}):= \left.\frac{d}{ds}L(f/K,\chi,s)\right|_{s=\frac{k+2}{2}}\neq 0.$$

This conjectural criterion relates non-vanishing of the constructed Stark–Heegner cycles to non-vanishing of the derivative of the twisted base-change $L$-function; the source gives no resolution.