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.