The de Rham–perverse-sheaf spectral sequence conjecture

Let $\mathrm{X}$ be a holomorphic symplectic manifold, and let $\mathscr{D}_\mathrm{L}$ and $\mathscr{D}_\mathrm{M}$ be quantised orientation modules associated to Lagrangian submanifolds $\mathrm{L}$ and $\mathrm{M}$. Let $\mathscr{DR}^{\mathrm{vir}}(\mathscr{D}_\mathrm{L},\mathscr{D}_\mathrm{M})$ denote their virtual de Rham complex, and let $\mathscr{P}_{\mathrm{L}\cap\mathrm{M}}[-n]$ be the shifted perverse-sheaf object. Spectral-sequence conjecture. There should be a simply graded spectral sequence whose first page is

$$\mathrm{E}_1^\bullet=\mathscr{DR}^{\mathrm{vir}}(\mathscr{D}_\mathrm{L},\mathscr{D}_\mathrm{M})$$

and which converges to

$$\mathrm{H}\mathscr{P}_{\mathrm{L}\cap\mathrm{M}}[-n].$$

This is motivated by the known quasi-isomorphism in the smooth-intersection case and by related conjectures for virtual de Rham complexes and perverse sheaves; the general-intersection case remains open.