nt.number theory – Asymptotic of rad(abc) in the abc conjecture

The abc conjecture famously predicts that, given any $epsilon>0$, for all but finitely many positive coprime integers $a,b,c$ with $a+b=c$, the radical $rad(abc)$ (i.e., the product of all prime factors of $abc$ without multiplicity) is at least $c^{1-epsilon}$. My question is, are there any quantitative predictions about what happens if one increases the exponent from $1-epsilon$ to something larger than $1$? Obviously, there are then examples such that the statement in the above form is not true any more, but how frequent are these? To be explicit, given any $d in (1,2)$ ($d>2$ might start to be nonsensical, since $abc$ would often be no larger than $O(c^2)$), what is a reasonable expected upper bound for the asymptotic proportion (depending on $N$) of coprime triples $(a,b,c)$, with $a+b=cle N$ such that $rad(abc) < c^{d-epsilon}$?

abc conjecture predicts $O(1)$ for $d=1$, but is there any guess for an asymptotic upper bound $N^{f(d)}$, with $f:(1,2)to (0,2)$?