The height bound conjecture for rational points on hyperelliptic curves

For an integer $d\geq 3$, let $f\in\mathbb{Q}[x]$ be a polynomial of degree $d$ with nonzero discriminant, and let $h(f)$ denote the maximum of the logarithmic heights of its coefficients. The height bound conjecture. There exist constants $C_1=C_1(d)$ and $C_2=C_2(d)$ such that, for every such polynomial $f$ and all $x,y\in\mathbb{Q}$ satisfying

$$t^2=f(x),$$

one has

$$h(x)\leq C_1\cdot h(f)+C_2.$$

The source presents this as a conjecture that, in degree at least $5$, follows from Vojta's conjecture over $\mathbb{Q}$; it is used to obtain infinitely many rational specializations with prescribed index-two arboreal Galois images.