Classical abc conjecture for the algebraic radical

For a non-zero integer $a$, let $N(a)$ be its algebraic radical,

$$N(a)=\prod_{p\mid a}p.$$

Classical abc conjecture. For every $\varepsilon>0$, there is a constant $K(\varepsilon)$ depending only on $\varepsilon$ such that whenever $a$, $b$, and $c$ are three coprime non-zero integers satisfying $a+b=c$,

$$c<K(\varepsilon)N(abc)^{1+\varepsilon}.$$

The conjecture is a foundational statement about additive relations among coprime integers and is invoked in the source to derive finiteness results for double-factorial equations. Its unconditional status is not resolved in the supplied text.