Shelah's eventual categoricity conjecture for abstract elementary classes

Let $\theta$ be a function assigning a cardinal $\theta(\tau)$ to each cardinal $\tau$. If $d4e6$ is an abstract elementary class categorical in some $d706 d5f9 d706(\operatorname{LS}(d4e6))$, then it is categorical in every $d706' d5f9 d706(\operatorname{LS}(d4e6))$. Shelah's eventual categoricity conjecture. There exists a function $d706 \mapsto \lambda(d706)$ such that if $d4e6$ is an AEC categorical in some $d706 \geq \lambda(\operatorname{LS}(d4e6))$, then $d4e6$ is categorical in all $d706' \geq \lambda(\operatorname{LS}(d4e6))$. This is a central categoricity problem for tame abstract elementary classes and asks for eventual upward transfer of categoricity; its resolution is not supplied here.