Lang's lower-bound conjecture for canonical heights on elliptic curves

Let $k$ be a number field, let $E/k$ be an elliptic curve, and let $P\in E(k)$ be a point of infinite order. Write $\widehat{h}$ for the Néron–Tate height on $E$, $\Delta_E$ for the minimal discriminant, and $j_E$ for the modular invariant. Then $\widehat{h}(P)$ should satisfy

Lang's conjecture. There is a positive constant $c(k)$, depending only on $k$, such that

$$\widehat{h}(P) \geq c(k)\,\max\Big\{\log \operatorname{N}_{k/\mathbb{Q}}(\Delta_E),h(j_E)\Big\}.$$

This is the elliptic-curve case of the Lang–Silverman lower-bound conjecture. The source explains that it is known in several partial cases, including under a uniformly bounded Szpiro quotient, but the general assertion remains open.