The Kähler contraction conjecture for null loci

Let $X$ be a compact Kähler manifold and let $[\alpha]$ be a nef and big $(1,1)$ class which is not Kähler, with $[\alpha]+\lambda c_1(X)$ a Kähler class for some $\lambda>0$. Let $\operatorname{Null}(\alpha)$ denote the null locus of $[\alpha]$. Kähler contraction conjecture. There is a bimeromorphic morphism $\pi:X\to Y$ onto a normal Kähler space $Y$ such that

$$\operatorname{Exc}(\pi)=\operatorname{Null}(\alpha)$$

and $[\alpha]=\pi^*[\omega_Y]$ for some Kähler class $[\omega_Y]$ on $Y$. The source calls this stronger statement true in the projective case and notes that it is easy in dimension two, with partial results in dimension three.