Lang's conjecture on canonical heights of elliptic-curve points

Let $E/\mathbb{Q}$ be an elliptic curve with $j$-invariant $j_E$ and minimal discriminant $\Delta_E$. Let $\hat h$ denote the canonical height and $h$ the relevant height on rational numbers. Lang's conjecture. There exists an absolute constant $c_L$ such that

$$\hat{h}(P) \geq c_{L}\max\{h(j_E),h(\Delta_E)\}$$

for all non-torsion points $P\in E(\mathbb{Q})$. This is a lower-bound conjecture for canonical heights and is used in the paper to establish Bremner's conjecture conditionally; its general status remains open.