Camere's conjecture on fixed loci of symplectic involutions on K3^[2]-type manifolds

Let $X$ be a hyperkähler manifold deformation equivalent to the Hilbert square of a $K3$ surface, and let $\varphi$ be an involution of $X$ preserving the holomorphic symplectic form. Denote its fixed locus by $X^{\varphi}$. Camere's conjecture. The fixed locus $X^{\varphi}$ does not contain complex tori. The theorem preceding this conjecture proves it by showing that every symplectic involution on a manifold of $K3^{[2]}$-type is deformation equivalent, as an automorphism, to one induced by a symplectic involution on a $K3$ surface; hence the conjecture is solved.