Automorphic Goldfeld conjecture for quadratic twists

Let $X$ be the set of quadratic fields, and let $\pi'$ be an irreducible cuspidal automorphic representation of $\operatorname{PGL}_2(\mathbb{A})$. For $E\in X$, let $\eta_E$ be the quadratic character attached to $E$. Automorphic Goldfeld conjecture.

The central value $L(\tfrac12,\pi'\otimes\eta_E)$ should be non-zero for half of the quadratic fields, namely

$$\lim_{x\to\infty}\frac{\#\{E\in X\mid |\Delta_E|<x,\ L(\tfrac12,\pi'\otimes\eta_E)\neq0\}}{\#\{E\in X\mid |\Delta_E|<x\}}=\frac12,$$

and it should be non-zero for a positive proportion of them, namely

$$\#\{E\in X\mid |\Delta_E|<x,\ L(\tfrac12,\pi'\otimes\eta_E)\neq0\}\gg x$$

when $x\to\infty$. These statements generalize Goldfeld's conjecture from elliptic-curve $L$-functions to cuspidal automorphic representations of $\operatorname{PGL}_2$, and are open in general.