The $abcd$-conjecture

Let $K$ be a number field or a one-dimensional function field of characteristic zero, and let $n\ge 3$. Let $[Z_1:\cdots:Z_n]$ be the standard homogeneous coordinates on $\mathbb{P}^{n-1}(K)$, and let $\mathcal{H}$ be the hyperplane given by

$$Z_1+\dots+Z_n=0.$$

For $P=(z_1,\dots,z_n)\in\mathbb{P}^{n-1}(K)$ with nonzero coordinates, let $h(P)$ be the height and $\operatorname{rad}(P)$ the radical defined in the source. The $abcd$-conjecture. For any $\epsilon>0$, there is a proper Zariski closed subset $\mathcal{Z}=\mathcal{Z}(K,\epsilon,n)\subsetneq\mathcal{H}$ and a constant $C_{K,\mathcal{Z},\epsilon,n}$ such that for all $P=(z_1,\dots,z_n)\in\mathcal{H}\setminus\mathcal{Z}$ with $z_1,\dots,z_n\in K^*$, we have

$$h(P)<(1+\epsilon)\operatorname{rad}(P)+C_{K,\mathcal{Z},\epsilon,n}.$$

This generalizes the $abc$-conjecture, which corresponds to $n=3$, and is used in the source to prove results on uniform boundedness and dynamical Lang-type statements. The source gives no resolution status for this formulation.