Goldfeld's conjecture for quadratic twists of automorphic representations

Let $F$ be a number field, let $\pi$ be a self-contragredient cuspidal automorphic representation of $\operatorname{GL}_2(\mathbb{A}_F)$, and let $\chi$ be a quadratic character with $\varepsilon(\pi\otimes\chi)=\varepsilon$. Fix a finite set $S$ of places of $F$.

Automorphic Goldfeld conjecture. Among quadratic characters $\chi'$ satisfying

$$\varepsilon(\pi\otimes\chi')=\varepsilon,\qquad \chi'_v=\chi_v\quad(v\in S),$$

the density of those for which $\operatorname{ord}_{s=1/2}L(s,\pi\otimes\chi')=0$ is one when $\varepsilon=+1$, and the density of those for which it equals $1$ is one when $\varepsilon=-1$. This generalizes Goldfeld's minimalist prediction from elliptic-curve twists to self-contragredient automorphic representations. The source gives no resolution status.