Local solubility of generalised Fermat equations

For every $n\ge 2$ we determine the asymptotic formula for the number of integer triples $(a,b,c)$ of bounded absolute value such that the generalised Fermat equation given by $a{x}^n+b{y}^n+c{z}^n=0$ is everywhere locally soluble. We compute the leading constant, answering a question of Loughran–Rome–Sofos, and determine that the conjectures of Loughran–Smeets and Loughran–Rome–Sofos hold for such equations.