The Vojta–Hall–Lang height bound for integral points on hyperelliptic curves

Let $d\geq 3$, let $f\in\mathbb{Z}[x]$ have degree $d$ and nonzero discriminant, and let $x,y\in\mathbb{Z}$ satisfy

$$y^2=f(x).$$

Vojta–Hall–Lang conjecture. There exist constants $C_1=C_1(d)$ and $C_2=C_2(d)$ such that

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

This is the conjectural height estimate used to obtain uniform control of integral points in the dynamical argument; the supplied text gives no resolution status or explicit values for the constants.