# What are the consequences of allowing the ABC-conjecture $\kappa_{\epsilon}$ to also vary with $\omega(abc)$?

A commonly encountered form of the ABC-conjecture is the following:

For all $\epsilon > 0$, there is a constant $\kappa_{\epsilon} > 0$ (depending only on $\epsilon$) such that for all coprime integers $a$, $b$ with sum $c = a + b$, \begin{eqnarray} \text{rad}(abc)^{1 + \epsilon} > \kappa_{\epsilon} \max ( |a|, |b|, |c| ). \end{eqnarray}

Related to his work on linear forms of logarithms, Baker conjectured the inequality:

\begin{eqnarray} (\epsilon^{-\omega(abc)} \text{rad}(abc))^{1 + \epsilon} > \kappa_{\epsilon} \max ( |a|, |b|, |c| ). \end{eqnarray}

What are the consequences, if any, for allowing $\kappa_{\epsilon}$ to no longer depend solely on $\epsilon$ but also have a dependence on $\omega(abc)$ (the number of distinct prime divisors of $abc$) in the original form of the conjecture? That is, for example, somewhere in between the two conjectures:

For all $\epsilon > 0$, there is a constant $\kappa_{\epsilon, n} > 0$ (depending only on $\epsilon$ and $n = \omega(abc)$) such that for all coprime integers $a$, $b$ with sum $c = a + b$, \begin{eqnarray} \text{rad}(abc)^{1 + \epsilon} > \kappa_{\epsilon, n} \max ( |a|, |b|, |c| ). \end{eqnarray}

Question: Are there any neat and non-trivial consequences preserved with this latter form of the conjecture, i.e., finite solutions of certain Diophantine equations, asymptotic bounds on linear forms for logarithms, etc.?