Le Boudec's conjecture on small points in quadratic twist families

Fix $F(x)=x^3+Ax+B\in\mathbb{Z}[x]$ with discriminant $\Delta=-(4A^3+27B^2)\neq0$, and let $E$ be the elliptic curve $y^2=F(x)$ over $\mathbb{Q}$. Let $\mathcal{S}(X)$ be the positive square-free integers up to $X$, let $E_d$ be the quadratic twist $dy^2=F(x)$ for $d\in\mathcal{S}(X)$, and define $\eta_d(A,B)$ by

\log\eta_d(A,B)=\min\left\\{\hat h_{E_d}(P):P\in E_d(\mathbb{Q})\setminus E_d(\mathbb{Q})_{\operatorname{tors}}\right\\}

when $\operatorname{rank}E_d(\mathbb{Q})\geq1$, with $\eta_d(A,B)=\infty$ otherwise. Define

\mathcal{N}_\alpha(A,B;X)=\\#\left\\{d\in\mathcal{S}(X):\eta_d(A,B)\leq d^{1/8+\alpha}\right\\},

and let $\lambda_{A,B}$ be the number of irreducible factors of $F(x)$ in $\mathbb{Z}[x]$.

Le Boudec's conjecture. For every $\alpha>0$, there exists a constant $c_{A,B}(\alpha)>0$ such that

$$\mathcal{N}_\alpha(A,B;X)\sim c_{A,B}(\alpha)X^{1/2}(\log X)^{\lambda_{A,B}}$$

as $X\to\infty$.

This is presented as the direct elliptic-curve analogue of Hooley's conjecture, with canonical heights of quadratic twists playing the role of fundamental units. The source says the conjecture was communicated by Le Boudec in private conversations and gives no evidence of a resolution.