Shen–Yin's perverse-Hodge symmetry conjecture

Let $\pi\colon M\to B$ be the Lagrangian fibration under consideration, and let

$$G_{i,k}=\operatorname{gr}_{-k}^F\operatorname{DR}(\mathcal{P}_i)[-i]$$

be the associated complexes of coherent $\mathscr{O}_B$-modules, where $i$ records the cohomological degree and $k$ the holomorphic degree. Shen–Yin's perverse-Hodge symmetry conjecture. In the derived category of coherent $\mathscr{O}_B$-modules, one has

$$G_{i,k}\cong G_{k,i}.$$

The two complexes are already isomorphic over the smooth locus of $\pi$; the conjecture asserts that this symmetry extends across the singular locus, where the behavior of the complexes is otherwise mysterious.