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The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski,predicts a correspondence between weight and perverse filtrations in thecontext of mirror symmetry. In this paper, we offer to reexamine thisconjecture through the lens of mirror symmetry for a Fano pair $(X,D)$ where$X$ is a smooth Fano variety, and $D$ is a simple normal crossing divisor. Amirror object is a multi-potential version of the Landau-Ginzburg (LG) model,called hybrid LG model, which encodes the mirrors of all irreducible componentsof $D$. We explain how the mirror P=W conjecture is induced by the relativeversion of homological mirror symmetry for such pairs. The key idea is to finda combinatorial description of the perverse filtration. The requisitecombinatorics naturally appears in the course of studying Hodge numbers ofhybrid LG models. We discuss this topic in the second half of the paper and useit to extend the work of Katzarkov-Kontsevich-Pantev to the hybrid LG setting.In application, we discover an interesting upper bound of the multiplicativityof the perverse filtration, which is an independent interest.

Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence