The core-map decomposition conjecture

Let $X$ be a compact Kähler manifold of dimension $n$, and let $c_X:X\dashrightarrow C(X)$ be its core map, whose general fibre is special and whose orbifold base is of general type. Let $J$ and $r$ be, respectively, the orbifold versions of the Moishezon fibration and the rational quotient. Write $\kappa_+$ for the relevant orbifold Kodaira dimension invariant.

Core-map decomposition conjecture. For any such $X$,

$$c_X=(J\circ r)^n.$$

In particular, special manifolds are towers of fibrations whose general fibres have either $\kappa=0$ or $\kappa_+=-\infty$.

The source states that this conjecture holds if the orbifold version of Iitaka's $C_{n,m}$-conjecture is true, so it is conditional on that conjecture in the paper. No unconditional resolution is given.