In a question I made few years ago, asking if $\frac{a^n+b^n}{a+b} = c^n$ can have a solution for $a,b,c,n\in \mathbb N$, $a^n+b^n\neq a+b$, and $\gcd(a+b,c)=\gcd(a,b)=1$, the user @MummytheTurkey conjectured that there were no non-trivial solutions for $n>3$, mentioning an heuristic of why the conjecture might be true (citation follows):
I only thought about the case when $n$ is odd. In that case the LHS is a polynomial of degree $n−1$. This equation then cuts out a curve in $\mathbb{P}(n,n,n−1)$ (weighted projective space). Solutions should conincide(ish) with rational points on this curve (call it $C_n$). When $n\geq 5$ this curve has genus $\geq 2$, so has only finitely many rational points by Faltings' Theorem. In practice when one searches for points on curves of genus $\geq 2$ with nice enough equations you usually find all the solutions straight away (at least if you use clever enough point searching)
I would like to know which strategy (if any) could be followed to show that this conjecture is indeed true (or false, for instance a strategy to look for a counterexample).
Thanks for your time!