The asymptotic Fermat conjecture over number fields

Let $K$ be a number field. A solution $(x,y,z)\in K^3$ of the Fermat equation

$$x^p+y^p+z^p=0$$

is trivial if $xyz=0$ and non-trivial otherwise. The asymptotic Fermat conjecture. If $\zeta_3\notin K$, then there exists a constant $\mathcal{B}_K$, depending only on $K$, such that for every prime $p>\mathcal{B}_K$, all solutions in $K^3$ are trivial. This conjecture predicts that, over number fields not containing a primitive third root of unity, non-trivial Fermat solutions occur only for bounded prime exponents; it is known for various classes of fields but remains open in general.