The eventual categoricity conjecture for $cmathbb{L}_{comega_1,comega}$

Let $cpsi$ be an $cmathbb{L}_{comega_1,comega}$-sentence. A sentence is categorical in a cardinal $cmu$ if it has exactly one model of cardinality $cmu$ up to isomorphism.

Eventual categoricity conjecture. There is a threshold cardinal $ctheta$ such that if $cpsi$ is categorical in some $cmu cge ctheta$, then $cpsi$ is categorical in all $cmu' cge ctheta$.

This captures the eventual content of the strong categoricity conjecture without fixing the threshold in advance. It remains open in ZFC, although the paper proves corresponding results from strong compactness assumptions.