Shelah's categoricity conjecture

Statement

Shelah introduced AECs to provide a uniform framework in which to generalize first-order classification theory. Classification theory started with Morley's categoricity theorem, so it is natural to ask whether a similar result holds in AECs. This is Shelah's eventual categoricity conjecture. It states that there should be a Hanf number for categoricity:

For every AEC K there should be a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ such that if K is categorical in some $\lambda \geq \mu$ (i.e. K has exactly one (up to isomorphism) model of size $\lambda$), then K is categorical in $\theta$ for all $\theta \ge \mu$.

Shelah also has several stronger conjectures: The threshold cardinal for categoricity is the Hanf number of pseudoelementary classes in a language of cardinality LS(K). More specifically when the class is in a countable language and axiomatizable by an $L_{\omega_1,\omega}$ sentence the threshold number for categoricity is $\beth_{\omega_1}$. This conjecture dates back to 1976.

Several approximations have been published (see for example the results section below), assuming set-theoretic assumptions (such as the existence of large cardinals or variations of the generalized continuum hypothesis), or model-theoretic assumptions (such as amalgamation or tameness). As of 2014, the original conjecture remains open.