Cohn's conjecture on the exponent in the equation $a^n+b^n=z^2$

Let $a$ and $b$ be integers with $2\leq a<b$, and consider the equation

$$a^n+b^n=z^2.$$

Here $(z,n)$ denotes a solution of the equation. Cohn's conjecture. Every solution $(z,n)$ satisfies

$$n\leq 4.$$

The conjecture was proposed by J. H. E. Cohn in 2002 and concerns a uniform upper bound on the exponent in this generalized Diophantine equation. The source does not provide evidence resolving the conjecture.