Kähler package conjecture for corrected perverse objects of conifold degenerations

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. Suppose there exists a polarizable mixed Hodge module

$$\mathcal P^H\in MHM(X_0)$$

whose underlying rational perverse sheaf is $\mathcal P$. Kähler-package conjecture. 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. The expected result extends the Hodge-theoretic properties of intersection cohomology to the corrected perverse object; the source states that these properties remain conjectural because a mixed-Hodge-module realization of $\mathcal P$ has not been established.