A website ( http://www.math.unicaen.fr/~nitaj/abc.html#Consequences ) says that the $abc$ conjecture implies that there are only finitely many solutions to the equation $x^n+y^n=z^n$ with $\gcd(x,y,z)=1$ and $n\ge 4$. This one I have proven.

Lang's Algebra (p. 196) says that the $abc$ conjecture implies that for all $n$ sufficiently large, there are no solutions to the equation $x^n+y^n=z^n$ with $\gcd(x,y,z)=1$ and $x,y,z\ne 0$. This one I have not proven (is it true?).

What are the strongest known assertions about Fermat's Last Theorem that follows from the $abc$ conjecture, and how are they proven? I searched the web but people tend to just say vague things like "implies asymptotic version of FLT" and such.