Holomorphic foliation associated with a semi-positive class of numerical dimension one

Let $X$ be a compact K\"ahler manifold and $\alpha$ be a class in the
Dolbeault cohomology class of bidegree $(1, 1)$ on $X$. When $\alpha$ admits at
least two smooth semi-positive representatives, we show the existence of a
family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic
foliation on a suitable domain of $X$ along whose leaves any semi-positive
representative of $\alpha$ is zero. As an application, we give the affirmative
answer to \cite[Conjecture 2.1]{K2019} on the relation between the
semi-positivity of the line bundle $[Y]$ and the analytic structure of a
neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.