Conjecture for the first two terms in the quadratic-twist vanishing ratio

Let $E$ be an elliptic curve of conductor $Q$, let $q\nmid Q$ be prime, and let $S^{\pm}(X)$ be the relevant families of fundamental discriminants with sign $\pm$. For $\lambda=\pm1$, set

$$S^{\pm}(X;q,\lambda)=\{d\in S^{\pm}(X):\chi_d(q)=\lambda\}$$

and define

$$R_q^{\pm}(X)=\frac{\sum_{d\in S^{\pm}(X;q,1),\ L_E(1,\chi_d)=0}1}{\sum_{d\in S^{\pm}(X;q,-1),\ L_E(1,\chi_d)=0}1}.$$

Also let

$$R_q=\left(\frac{q+1-a_q}{q+1+a_q}\right)^{1/2},$$

where $a_q$ is the $q$th coefficient of the $L$-function of $E$. Conjecture for the first two terms. For $q\nmid Q$,

$$R_q^{\pm}(X)=R_q\frac{1+\frac{3}{8\log(X)}(\beta^{\pm}_{-\frac12}(q,1)-1)}{1+\frac{3}{8\log(X)}(\beta^{\pm}_{-\frac12}(q,-1)-1)}+O(\log(X)^{-2}),$$

where $\beta^{\pm}_{-\frac12}(q,\lambda)$ is given explicitly in the source, and the implied constant depends on $E$ and $q$, hence also on $a_q$. This refines the predicted limiting ratio by specifying its first correction in powers of $1/\log X$, based on conjectural moment asymptotics; no resolution is given in the supplied text.