The uniruledness conjecture for null-locus components

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 $\lambda>0$. The null locus $\operatorname{Null}(\alpha)$ is the locus associated with the nef and big class where it fails to be numerically positive. Uniruledness conjecture. Every irreducible component of $\operatorname{Null}(\alpha)$ is uniruled. The conjecture is known in the projective case under the stated numerical setting and in complex dimension two, but remains open in general Kähler geometry.