By Saito's theory it is satisfied for analytic varieties. In a joint work with J-P. Brasselet we have constructed in canonical way Chern classes in intersection homology. They are computable in very few cases. One should find their meaning and effective ways of computation. I have managed to prove this duality in pure algebraic context, when a reductive Lie algebra g acts on differential graded module.
Further generalizations are possible for super Lie algebras with differential.
The appropriate language for this problem is Hohschield cohomology, deformation theory, operads. This is joint work with Marcin Chalupnik and Tomasz Maszczyk. Such a form may give element in local intersection homology. The problem of lift of Leray residue to intersection homology may be rephrased in the language of D-modules, and it seems that in this setup one should find a general answer.