Le Boudec's height lower-bound conjecture for quadratic twists

For integers $A,B$ with $4A^3+27B^2\neq 0$, consider the quadratic twists

$$dy^2=x^3+Ax+B$$

for positive square-free integers $d$, and let $\eta_d(A,B)$ be defined by

$$\log \eta_d(A,B)=\min\{\hat{h}_{E^{(d)}}(P):P\in E^{(d)}(\mathbb{Q})\setminus E^{(d)}(\mathbb{Q})_{\mathrm{tors}}\},$$

with $\eta_d(A,B)=+\infty$ if the set is empty. Write $S(X)$ for the set of positive square-free integers at most $X$. Le Boudec's conjecture. For fixed $\epsilon>0$, the set of square-free $d\in\mathbb{Z}_{\geq 1}$ such that

$$\eta_d(A,B)>e^{d^{1/2}-\epsilon}$$

has natural density $1$ in $S(X)$. This predicts that, for almost all quadratic twists in the family, the least nonzero canonical height is at least approximately $d^{1/2}$ on the logarithmic scale.