Alessandrini–Bassanelli conjecture on the hierarchy of $p$-Kähler structures

Let $(M,J)$ be a complex manifold. A $p$-Kähler structure on $(M,J)$ is a closed, pointwise transverse $(p,p)$-form. Alessandrini–Bassanelli conjecture. If $(M,J)$ admits a $p$-Kähler structure, then it admits $q$-Kähler structures for all $q\ge p$. This conjecture asks whether the existence of a $p$-Kähler structure implies the existence of all higher-degree structures in the $p$-Kähler hierarchy. The paper proves it for nilmanifolds with nilpotent complex structures and for holomorphically parallelizable solvmanifolds, while the general question remains open.