The abc conjecture for generalized Fermat equations with variable coefficients

Let $a,b,c$ be nonzero integers, and let $(x,y,z,p,q,r)$ be integers satisfying the generalized Fermat equation $eqref{eq:genfermat}$, with $\gcd(x,y,z)=1$, $p,q,r>0$, and $1/p+1/q+1/r<1$. Solutions for which $\min\{x,y,z\}=1$ and that differ only in the exponents of $1$ are identified. Generalized Fermat abc conjecture. For any integers $a,b,c$ with $abc\neq0$, there are only finitely many such solutions in $(x,y,z,p,q,r)$. This conjecture is presented as a consequence expected from the abc conjecture and extends the Fermat–Catalan finiteness perspective to fixed nonzero coefficients. Its status is not specified in the supplied text.