Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class C if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki.
Properties.
Let "M" be a compact manifold of Fujiki class C, and 
formula_1 its complex subvariety. Then "X"
is also in Fujiki class C (, Lemma 4.6). Moreover, the Douady space of "X" (that is, the moduli of deformations of a subvariety formula_1, "M" fixed) is compact and in Fujiki class C.
Conjectures.
J.-P. Demailly and M. Pǎun have
shown that a manifold is in Fujiki class C if and only
if it supports a Kähler current. 
They also conjectured that a manifold "M" is in Fujiki class C if it admits a nef current which is "big", that is, satisfies
For a cohomology class formula_4 which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle "L" with first Chern class
nef and big has maximal Kodaira dimension, hence the corresponding rational map to
is generically finite onto its image, which is algebraic, and therefore Kähler.
Fujiki and Ueno asked whether the property C is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun 