 Solution

The theory is not categorical in any infinite cardinal. In cardinality ${\aleph }_0$, one may add no infinite equivalence class or one countably infinite class; these models are not isomorphic. For uncountable $\kappa$, one can vary the number and cardinalities of infinite classes while retaining infinitely many classes of every finite size. These choices are isomorphism invariants.

Solved by gpt-5.6-sol high.