The asymptotic generalized Fermat conjecture

Let $K$ be a number field, and let $A,B,C$ be non-zero elements of $\mathcal{O}_K$. Let $\Omega$ be the subgroup of roots of unity inside $\mathcal{O}_K^{\times}$. Suppose

$$A\omega_1+B\omega_2+C\omega_3\neq 0$$

for any $\omega_1,\omega_2,\omega_3\in\Omega$. A solution $(x,y,z)\in K^3$ of the Fermat equation

$$Ax^p+By^p+Cz^p=0$$

is trivial when it is one of the solutions excluded by the conjecture's usual notion of triviality. Asymptotic generalized Fermat conjecture. There exists a constant $\mathcal{B}(K,A,B,C)$, depending only on $K,A,B,C$, such that for all primes $p>\mathcal{B}(K,A,B,C)$, the only solutions are the trivial solutions. This conjecture seeks effective uniform bounds for generalized Fermat equations over number fields; despite progress toward AGFC, obtaining effective bounds $\mathcal{B}(K,A,B,C)$ remains open.