The simple-manifold decomposition conjecture

Let $X_n$ be a simple compact Kähler manifold of dimension $n$. A manifold is Kummer if it is bimeromorphic to a quotient of a complex compact torus by a finite group, and bimeromorphically irreducible symplectic if it is bimeromorphically symplectic with $h^{2,0}=1$.

Simple-manifold decomposition conjecture. $X_n$ is either bimeromorphically irreducible symplectic or Kummer. In particular, $X_n$ is Kummer if $n$ is odd.

This conjecture is used to identify simple compact Kähler manifolds in the conditional description of manifolds of algebraic dimension zero. Its status is not specified in the source.