The Conway-group classification conjecture for symplectic automorphisms of manifolds of $K3^{[n]}$ type

Let $X$ be a manifold obtained as a smooth deformation of the Hilbert scheme of $n$ points on a $K3$ surface, called a manifold of $K3^{[n]}$ type. Let $Co_1$ be the Conway group, and let $G$ be a finite group of symplectic automorphisms of such a manifold for some $n$. The subgroup conditions referred to are those in the preceding sufficient proposition: $G$ is a subgroup of $Co_0$ whose invariant sublattice in the Leech lattice has rank at least $4$ and whose discriminant group has fewer generators than its rank.

Conway-group classification conjecture. There is a bijective correspondence between finite groups of symplectic automorphisms of manifolds of $K3^{[n]}$ type (for some $n$) and subgroups $G$ of $Co_1$ satisfying these conditions.

This conjecture seeks a complete classification of finite symplectic automorphism groups in the $K3^{[n]}$-type setting, extending the realization of symplectic automorphism groups on $K3$ surfaces through the Conway group. The paper presents the claim as a conjecture and gives sufficient realization results, but no resolution is supplied here.