Decomposition conjecture for compact Fujiki manifolds with vanishing first Chern class

Let $X$ be a compact Fujiki manifold such that $c_1(X)=0\\,\in\,H^2(X,\mathbb{R})$. A finite étale cover of $X$ is a finite étale morphism $X'\longrightarrow X$. An irreducible Calabi–Yau manifold and an irreducible holomorphic symplectic manifold are the factors specified below; an ICY variety is a variety with trivial canonical class in the intended sense of the source, and terminal singularities are understood in the standard birational-geometric sense. The conjecture asserts that there is a finite étale cover $X'\longrightarrow X$ and a decomposition

$$X'\simeq T\times\prod_{i\in I}Y_i\times\prod_{j\in J}Z_j.$$

Here $T$ is a complex torus, each $Y_i$ is an irreducible Calabi–Yau manifold, and each $Z_j$ is an irreducible holomorphic symplectic manifold. Moreover, there are bimeromorphic maps $Y_i\dashrightarrow\widehat Y_i$ and $Z_j\dashrightarrow\widehat Z_j$, isomorphic in codimension one, such that each $\widehat Y_i$ is a projective ICY variety with terminal singularities and each $\widehat Z_j$ is a Kähler IHS variety with terminal singularities.

Decomposition Conjecture. Every compact Fujiki manifold with vanishing first Chern class admits such a finite étale cover and decomposition, together with the stated bimeromorphic models.

This proposes a Beauville–Bogomolov-type decomposition for compact Fujiki manifolds with vanishing first Chern class, extending the classical decomposition theorem beyond the Kähler setting. The parser supplies no resolution evidence, so the conjecture is recorded as open; the precise definitions of the Calabi–Yau, ICY, and IHS terminology should be checked against the paper.