The Diophantine equation $x^n - y^n = z^2$

Question

Darmon-Merel theorem (DMT) ensures that if $n \geq 4$ is an integer and $x, y, z > 0$ are integers such that $(x, y, z) = 1$ then $x^n + y^n \neq z^2.$

The question is: Does DMT apply to the equation $x^n - y^n = z^2$?