Local contractibility of compactly supported Hamiltonian diffeomorphism groups on open symplectic manifolds

Ascertain whether the compactly supported Hamiltonian diffeomorphism group Ham_c(M, ω), equipped with the subspace topology inherited from Diff_c(M), is locally contractible for arbitrary connected open symplectic manifolds (M, ω).

Background

For open symplectic manifolds, the authors equip Diff_c(M) with the colimit topology and consider the induced subspace topology on Symp_c(M, ω). While Symp_c(M, ω) is locally contractible, the status of local contractibility for Ham_c(M, ω) is unknown in general. This uncertainty is tied to the behavior of the flux group associated with the symplectomorphism group, which may fail to be discrete on general open manifolds.