Perverse coherent sheaves on hyperkahler manifolds and Weil conjectures

In Beilinson-Bernstein-Deligne (BBD), Weil conjectures were interpreted as a teorem about purity of a direct image of a pure perverse sheaf. Coherent sheaves on a general (non-algebraic) deformation of hyperkahler manifold have all singularities in codimension 2, which allows one to define a self-dial middle perversity on this category. However, the category of coherent sheaves on a general hyperkahler variety admits a full embedding to the category of coherent sheaves on any its deformation (and is essentially independent on the deformation), hence the notion of “perverse coherent sheaf” makes sense on algebraic hyperkahler manifolds as well.
Instead of fixing the Frobenius action, as in the BBD setup, one should fix the lifting of the sheaf to the twistor space. The role of weight filtration is played by the $O(i)$-filtration on the sheaf restricted to the rational curves in the twistor space (the corresponding filtration is actually the Harder-Narasimhan filtration on the sheaves over twistor spaces). The hyperkahler version of “Weil conjectures” predicts that the weights are increased under pushforwards, and the pushforwards of pure perverse sheaves remain pure, which is actually true, at least in the smooth case.