Let $E: y^2 = x^3 + Ax + B$ be a quasi-minimal elliptic curve over $\mathbb{Q}$, i.e. $\gcd(a^3, b^2)$ is $12$th power free. Furthermore, let $\operatorname{rank}(E) = 1$ and $j(E)=\frac{1728 \times 4A^3}{4A^3 + 27B^2} \in \mathbb{Z}$.

Given these conditions, is the subset of these elliptic curves conjectured or known to have a positive proportion among all elliptic curves over $\mathbb{Q}$? Initial heuristic approaches suggest no, but definitive insights or references would be highly appreciated.

Thanks!