The prescribed Hanf threshold in Shelah's categoricity conjecture is not provable in ZFC

Theorem 1.1 (assuming CH): there is an AEC $K$ in a countable finitary relational language with $LS(K)=\aleph_0$, at least two nonisomorphic models of cardinality $\beth_{\omega_2}=H(K)$, and categorical in every cardinal $\ge\Lambda=\beth_{(2^{\aleph_1})^+}$. So categoricity at $\Lambda^+$ does not transfer down to $H(K)$, and the prescribed-threshold form is not a theorem of ZFC (if ZFC is consistent). The same class is eventually categorical, so this is compatible with the qualitative conjecture. It does not decide whether the prescribed form can hold under other hypotheses (for example failure of CH), and it does not concern the $\beth_{\omega_1}$ threshold for $L_{\omega_1,\omega}$.

This result conflicts with a published theorem. The release's manuscript states that its construction contradicts Espindola's published Theorem 4.1, which claims categoricity transfer at this bound. Until the conflict is settled in public, the entry is Contested. Theorem 1.1 (assuming CH): there is an AEC $K$ in a countable finitary relational language with $LS(K)=\aleph_0$, at least two nonisomorphic models of cardinality $\beth_{\omega_2}=H(K)$, and categorical in every cardinal $\ge\Lambda=\beth_{(2^{\aleph_1})^+}$. So categoricity at $\Lambda^+$ does not transfer down to $H(K)$, and the prescribed-threshold form is not a theorem of ZFC (if ZFC is consistent). The same class is eventually categorical, so this is compatible with the qualitative conjecture. It does not decide whether the prescribed form can hold under other hypotheses (for example failure of CH), and it does not concern the $\beth_{\omega_1}$ threshold for $L_{\omega_1,\omega}$.