Lang–Silverman conjecture for elliptic curves

Let $K$ be a number field, let $E/K$ be an elliptic curve, and let $P\in E(K)$. Let $L=3(O)$, where $O$ is the origin of $E$, and write $\widehat{h}(P)=\widehat{h}_{E,L}(P)$ for the Néron–Tate height associated with $L$. Let $\mathop{h_{\mathrm{F}^+}}(E/K)$ denote the relative differential height of $E/K$. Lang–Silverman conjecture. There exists a positive constant $c_4=c_4(K)$ such that, whenever $\mathbb{Z}\!\cdot\!P$ is Zariski dense in $E$, one has

$$\widehat{h}(P)\geq c_4\,\max\left\{\mathop{h_{\mathrm{F}^+}}(E/K),1\right\}.$$

This is the dimension-one case of the Lang–Silverman conjecture, relating the height of a nontorsion point to the differential height of the elliptic curve. The statement is presented in the source as the original conjecture in dimension one; no resolution status is supplied here.