There are two Fermat-Catalan solutions that have as an eighth power in their addend the numbers, $33^8$ and $44^8$. In Darmon and Granville's paper, they show that the generalized Fermat Equation has only a finite number of solutions when $1/p+1/q+1/r<1$ and an infinite number of solutions when $1/p+1/q+1/r>1$.
Now, where does that leave us for the Fermat-Catalan solutions:
$33^8+1549034^2=15613^3$ and
$43^8+96222^3=30042907^2$ ??
In these cases, it would appear that $1/8+1/3+1/2 = 1/8+1/2+1/3 <1$. BUT,
$33^8=1185921^2$, so I could just as easily say that for this equation, $1/2+1/3+1/2>1$, and likewise for the other equation. When taken as a square, it belongs to a group of infinite solutions. When taken as an eighth power it is taken as part of a finite group of solutions. Which is it? Where should it be counted? How can you belong to an infinite group and a finite group at the same time, when the identity between the eighth power and the square indicates you are dealing with one unique solution!?
EDIT 1: The question is sharpened if we consider the result from Bennet-Chen that the signature $(2,6,n)$ has no solutions.
Since $y^6=(y^3)^2$ then is there a solution or not? Bennet-Chen proved there is no solution, but $(2,2,n)$ has an infinite number of solutions. One can only reconcile the two results if circles $(x,y^3,z^{(n/2)})$ don't exist in the infinite family of $(2,2,n)$ for pairwise coprime $x,y,z$ and integer $n$.