The Vojta conjecture for integral points on hyperelliptic curves

For each integer $d\ge 3$, let $f\in\mathbb{Z}[x]$ have degree $d$ and nonzero discriminant, and let $h(f)$ be the maximum of the logarithmic heights of the coefficients of $f$. Vojta's conjecture. There exist constants $C_1=C_1(d)$ and $C_2=C_2(d)$ such that, whenever $x,y\in\mathbb{Z}$ satisfy

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

then

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

This uniform height bound is invoked in the paper to control the indices at which the arboreal Galois extensions can fail to be maximal. The source treats it as an assumed conjectural input, so its status here is open.