Grossberg–VanDieren's categoricity-to-tameness conjecture for abstract elementary classes

Let $K$ be an abstract elementary class (AEC), let $LS(K)$ denote its Löwenheim–Skolem number, and let $Hanf(LS(K))$ be the relevant Hanf bound. A class is $\chi$-tame when types over models are distinguished by their restrictions to submodels of size at most $\chi$.

Grossberg–VanDieren's categoricity-to-tameness conjecture. Suppose $K$ is an AEC. If $K$ is categorical in some $\lambda \geq Hanf(LS(K))$ (or some other value depending only on $LS(K)$), then there exists $\chi < Hanf(LS(K))$ such that $K$ is $\chi$-tame.

The conjecture asks whether sufficiently high categoricity forces tameness with a bound depending only on the Löwenheim–Skolem number. The supplied text does not state whether it has been resolved, so its database status remains open.