A CH obstruction to a prescribed categoricity threshold
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ 0 that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models of cardinality $\beth_{\omega_2}$. Thus categoricity does not transfer down to the proposed bound $\beth_{(2^{\aleph_0})^+}$, which equals $\beth_{\omega_2}$ under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.
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