Uniform size-bound conjecture for irreducible solutions of $(E_N)$

Let $N$ be an integer greater than $2$, and let the size of an irreducible solution of $(E_N)$ mean the size notion defined earlier in the paper. Uniform size-bound conjecture. There exists a strictly positive integer $K$ such that, for every integer $N>2$, all irreducible solutions of $(E_N)$ have size less than $N+K$.

This is the second conjecture arising from the cases treated in the preceding section. The supplied text gives no evidence that the conjecture has been proved or disproved.