Suppose $F(x,y,z)$ is a homogeneous polynomial over $\mathbb{Q}$, where $C:F(x,y,z)=0$ is a curve of genus $g\geq 2$.

**Question:** Faltings proved that $C$ has finite many rational points. Suppose that there is no rational points on $C$. Is there any effective version of Mordell conjecture, i.e., $$\vert F(x,y,z)\vert\gg_{F,g} g(H(x,y,z)),(x,y,z)\in\mathbb{Z}^3$$
where $H$ is naive height and $g(x)$ tends to infinity as $x$ tends to infinity?