A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, where $\text{rad}(k)$ denotes the product of distinct prime divisors of $k$.
One version of the $abc$-Conjecture is that for each $\varepsilon > 0$, there are finitely many $abc$-triples such that $P(a,b,c) > 1 + \varepsilon$.
There are finitely many known triples satisfying $P > 1.4$, the so called good triples, and the largest (quality) is $P(2,3^{10} \cdot 109, 23^{5}) = 1.629911684 \dots$ (discovered by E. Reyssat).
Question: Is there an upper bound for $P(a,b,c)$ simply in terms of $c$ and absolute constants (sharper than $\log_{2} c$)?
Question: Is there an absolute upper bound for $P(a,b,c)$ so that no triple has higher quality?
Best Answer: If there were such a bound, then asymptotic FLT would be in hand. (Thanks Ace of Base on MO!)