Uniform boundedness of S-units in rational-function images

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all Archimedean places, and let $\operatorname{{\mathfrak o}}^*_S$ denote the group of $S$-units. For integers $s\geq 1$ and $d\geq 2$, consider a degree-$d$ rational function $\phi(z)\in K(z)$ that is not a $d$-th power in $\overline{K}(z)$. Uniform image-set conjecture. There is a constant $C=C(s,d)$ such that, whenever $|S|=s$ and $S\supseteq S_\infty$,

$$\left|\phi(K)\cap\operatorname{{\mathfrak o}}^*_S\right|\leq C.$$

Siegel's theorem gives finiteness for each fixed $(K,S,\phi)$, and the paper proves the bound for several classes of rational functions and derives the full conjecture from a variant of the Caporaso--Harris--Mazur conjecture; uniformity in $K$ and $\phi$ remains open.