Kähler package conjecture for corrected perverse cohomology

Let $\pi:\mathcal X\to\Delta$ be a projective conifold degeneration, and let $\mathcal P$ be the corrected perverse object constructed from nearby and vanishing cycles. A mixed-Hodge-module refinement of $\mathcal P$ means a refinement compatible with the projective geometry of the degeneration. Kähler package conjecture. If $\mathcal P$ admits such a refinement, then the hypercohomology groups

$$\mathbb H^k(X_0,\mathcal P)$$

should satisfy a Kähler-type package analogous to that of intersection cohomology, including duality, Lefschetz-type isomorphisms, and Hodge–Riemann bilinear relations. This extends the established Hodge-theoretic package for intersection cohomology to the corrected perverse object; the required mixed-Hodge-module realization and the resulting duality, Lefschetz, and Hodge–Riemann statements remain conjectural in this setting.