Weak Goldfeld positive-proportion conjecture

Let $E$ be a fixed elliptic curve over $\mathbb{Q}$ and let $E^{(d)}$ be its quadratic twists. For $X>0$, let

$$N_r(X):=\left|\left\{|d|<X\mid r_{an}(E^{(d)})=r\right\}\right|.$$

Weak Goldfeld conjecture. For each $r\in\{0,1\}$,

$$\liminf_{X\to\infty}\frac{N_r(X)}{\left|\{|d|<X\}\right|}>0.$$

This weaker form asks only for a positive proportion of twists of analytic rank zero and one, rather than the $50\%$ proportions in Goldfeld's conjecture; the supplied text presents it as unresolved in general.