On a conjecture of Kashiwara

Kashiwara conjectured that the hard Lefshetz theorem and the semisimplicity theorem hold for any semisimple perverse sheaf M on a variety over a field of characteristic 0. He also conjectured that if you apply to such M the nearby cycle functor corresponding to some function then the successive quotients of the monodromy filtration are semisimple. We prove that these conjectures would follow from de Jong's conjecture on representations modulo l of the arithmetic fundamental group of a variety over a finite field.