The H-refined abc conjecture

For $n\in\mathbb{N}$, let $\gamma(n)=\prod_{p\mid n}p$ and let $\omega(n)$ denote the number of distinct prime factors of $n$. Define

$$H(n):=\frac{\gamma(n)}{(\log\gamma(n))^{\omega(n)}}.$$

The convention $0^0=1$ is used when $n=1$. The H-refined abc conjecture. For every $\epsilon>0$, there exists a constant $C(\epsilon)$ such that for all $(a,b,c)\in\mathbb{N}^3$ with $\gcd(a,b,c)=1$ and $a+b=c$,

$$c<C(\epsilon)H(abc)^{1+\epsilon}.$$

The paper proposes this as a stronger abc-type conjecture because $H$ is intended to retain more arithmetic information than the radical. The source states applications and heuristics for it, but gives no resolution.