A conjecture on diophantine equation

Question

Let $A, B, C$ be given three integer numbers, $A^2+B^2+C^2 \ne 0$; and $f(x,y,z)=0$ be given diophantine equation.

Then exist $N_0 > 2$ such that $m, n, k \ge N_0$ the equation as follows:

$f(x,y,z)+Ax^n+By^m+Cz^k=0$ with condition $gcd(x,y)=gcd(y,z)=gcd(z,x)=1$ has only finitely solutions

Question: The conjecture above is true?

Answer 1

Try $f(x,y,z) = x - y$, $A = 1$, $B = -1$, $C = 0$, $m = n$.