Gross–Kohnen–Zagier-type conjecture for Darmon's points

Assume that $E(F)$ has rank $1$, choose a generator $P_0$ of $E(F)$ modulo torsion, and for each admissible totally positive $t\in O_F$ let $P_t\in E(F)$ be the trace of the Darmon point attached to $K[t]$. Write $P_t=[P_t]P_0$ modulo torsion. Gross–Kohnen–Zagier-type conjecture. There exists a Hilbert modular form $g$ of level $3/2$ such that the integers $[P_t]$ are proportional to some Fourier coefficients of $g$.

This predicts that traces of Darmon points are encoded by Fourier coefficients of a half-integral-weight Hilbert modular form, in the spirit of Gross–Kohnen–Zagier; the supplied source gives no resolution.