To fix notation and check that my definitions are correct I will first state:
abc conjecture: Let $a,b\in\mathbb{N}$ be coprime, $c:=a+b$ , and define the quality of the triple $(a,b,c)$ to equal $q(a,b,c):=\log(c)/\log(\text{rad}(abc))$. Then for each $\varepsilon>0$, the number of such $(a,b,c)$ with $q>1+\varepsilon$ is finite.
However, looking at this table on Wikipedia of qualities of triples with $c<10^{18}$, it would seem possible that for $q\in[1,3/2]$ we have that the number $N(q,n)$ of triples with quality $>q$ and $c<n$ satisfies $N(q,n)=\Omega_q(\log n)$.
This would contradict the abc conjecture, so is there a reason that this behaviour should eventually stop for large $c$? Are there any other reasons (aside from its consequences) to believe that it is true?
My understanding of the conjecture is very superficial - I understand (from Silverman's Arithmetic of elliptic curves, section VIII) that abc implies Szpiro's conjecture, and conversely Szpiro's conjecture implies abc with the 1 replaced with 3/2, but this is the extent of my knowledge. I would also be interested in reasons why one might believe Szpiro's conjecture, which I currently do not have any intuition about.
