Problem 24.4.34A65ADE22C9ED

Problem 24.4.34.A65ADE22C9ED Let $\theta $ be a smooth closed semipositive real $(1,1)$-form on a compact complex manifold $X$ whose kernel has constant rank. Suppose that every leaf of the null foliation of $\theta $ is Liouville, in the sense that plurisubharmonic functions bounded above on it are constant, and that the closure of every leaf is a compact real submanifold whose real codimension equals the complex rank of $\theta $. Is every $\theta $-plurisubharmonic function continuous? If not, describe the possible singularity types in $\mathrm{PSH}(X,\theta )$ in terms of the closures of the leaves.