The finitely generated subgroup recognition conjecture for compact manifolds

Let $M$ and $N$ be compact connected $C^p$ manifolds, and let $\Diff^p_c(M)_0$ and $\Diff^p_c(N)_0$ denote their groups of compactly supported $C^p$ diffeomorphisms diffeotopic to the identity through a compactly supported diffeotopy. The finitely generated subgroup recognition conjecture. $M$ and $N$ are $C^p$-diffeomorphic if and only if the sets of isomorphism classes of finitely generated subgroups of $\Diff^p_c(M)_0$ and $\Diff^p_c(N)_0$ coincide. This conjecture asks whether the collection of finitely generated subgroup types determines a compact connected manifold up to $C^p$-diffeomorphism; the supplied text gives no resolution.