Conrey–Keating–Rubinstein–Snaith conjecture for prime quadratic twists

For a newform $f$ and the cuspidal automorphic representation $\pi$ of $\operatorname{PGL}_2(\mathbb{A})$ generated by $f$, let $\Delta_E$ be the discriminant of a quadratic field $E$, and let $\eta_E$ be its associated quadratic character. Conrey–Keating–Rubinstein–Snaith conjecture. For weight $2$ with rational Fourier coefficients, there is a constant $c_f^\pm\geq 0$ such that

$$\#\left\{E\,\middle|\,|\Delta_E|\textup{: \prime},\,0<\pm\Delta_E<X,\,\varepsilon\left(\tfrac12,\pi\otimes\eta_E\right)=1,\,L\left(\tfrac12,\pi\otimes\eta_E\right)=0\right\}\sim c_f^\pm X^{3/4}(\log T)^{-5/8},$$

and for weight $4$ with rational Fourier coefficients, there is a constant $c_f^\pm\geq0$ such that the same count is asymptotic to $c_f^\pm X^{1/4}(\log T)^{-5/8}$. The weight-$2$ case includes newforms corresponding to elliptic curves over $\mathbb{Q}$. This predicts precise rates for vanishing central values among prime-discriminant quadratic twists; the source presents it as an open expectation and compares its results with the weight-$4$ case.