The weak abc-conjecture for a number field

Let $K$ be a number field, let $M_K$ be its set of places, and define the height $\operatorname{h}$ and conductor $\operatorname{cond}$ as in the preceding number-field abc formulation. Weak abc-conjecture. There exists $\kappa>1$ such that, for every $x\in K$ distinct from $0$ and $1$,

$$\operatorname{h}(x)\le \kappa\bigl(\operatorname{cond}(x)+\operatorname{cond}(x^{-1})+\operatorname{cond}(x-1)+1\bigr).$$

This is explicitly presented as a weaker version of the number-field abc-conjecture and is the form used in the paper.