Let X and Y be compact hyper-Kahler manifolds deformation equivalent to the Hilbert scheme of n points on a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subalgebra generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of the second rational cohomologies of X and Y with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic. The case n=1 (when X and Y are K3 surfaces) was known as the Shafarevich conjecture and was solved by Nikolay Buskin in 2015 using work of Mukai. The higher dimensional case is based on recent results on equivalences of derived categories of hyperkahler varieties due to Taelman. We will also comment on the analogous result for hyperkahler manifolds of generalized Kummer deformation type (work in progress).