Classification conjecture for compact Kähler manifolds without analytic subvarieties

Let $X$ be a compact Kähler manifold which does not contain any nontrivial analytic subvarieties.

Classification conjecture. The manifold $X$ should be either a complex torus or an irreducible hyperkähler manifold. In particular, $K_X$ should be trivial, and the two cases should be distinguished by whether $q(X)>0$ or $q(X)=0$.

This is a more rigid form of the conjectural classification of simple compact Kähler manifolds. The source presents it as a conjectural picture, and no resolution of this statement is given.