Finiteness conjecture for irreducible solutions of $(E_N)$

Let $N\in\mathbb{N}^{*}$ with $N\geq 2$. A solution of $(E_N)$ is called irreducible when it cannot be decomposed as a sum of two nontrivial solutions, as in the preceding discussion. Finiteness conjecture. The equation $(E_N)$ has only finitely many irreducible solutions.

This is the first of two conjectures motivated by the cases treated in the paper. The supplied text does not state whether it has been resolved.