The finiteness conjecture for topological types of compact hyper-Kähler manifolds

For each fixed complex dimension, consider compact hyper-Kähler manifolds and identify two such manifolds when they are homeomorphic, equivalently when they have the same topological type. The topological finiteness conjecture. There are only finitely many topological types of compact hyper-Kähler manifolds in each dimension. The paper presents this as a consequence expected from the existence of SYZ fibrations and does not establish it.