Density-zero conjecture for central vanishing of elliptic-curve twists

Let $p>2$ be a prime, let $E/{\mathbb Q}$ be an elliptic curve, and let $\Sigma_p$ denote the set of Dirichlet characters of order $p$. For $S\subset\Sigma_p$, say that $S$ has density $\alpha$ if

$$\lim_{x\to\infty} \frac{\#\{\chi\in S\mid N(\chi)<x\}}{\#\{\chi\in\Sigma_p\mid N(\chi)<x\}}=\alpha,$$

where $N(\chi)$ is the conductor of $\chi$. Density-zero conjecture. For every elliptic curve $E/{\mathbb Q}$, the characters $\chi\in\Sigma_p$ for which $L(E,\chi,1)=0$ have density $0$ in $\Sigma_p$. This is a standard conjectural input concerning analytic ranks of elliptic-curve twists, used in the paper to support expectations about the Mordell–Weil rank modulo $n$ not being expressible as a sum of local terms for $n>2$.