Height lower-bound conjecture for sextic twists of Mordell curves

For the Mordell curves

$$E_d:y^2=x^3+d,$$

let $\zeta_d$ be defined by

$$\log \zeta_d=\min\{\hat{h}_{E_d}(P):P\in E_d(\mathbb{Q})\setminus E_d(\mathbb{Q})_{\mathrm{tors}}\},$$

with $\zeta_d=+\infty$ if the set is empty. Let

$$S_6(X)=\{d\in\mathbb{Z}:|d|\leq X,\ d\text{ is sixth-power free}\}.$$

The sextic-twist height conjecture. For fixed $\epsilon>0$, the set of $d\in S_6(X)\setminus\{0\}$ such that

$$\zeta_d>e^{|d|^{1/6}-\epsilon}$$

has natural density $1$ in $S_6(X)$. Thus, for almost all nonzero sixth-power-free parameters, the least nonzero canonical height is at least approximately $|d|^{1/6}$ on the logarithmic scale.