The proper subgroup conjecture for Hamiltonian homeomorphisms

Let $Hameo(M,\omega)$ be the group of Hamiltonian homeomorphisms and let $Sympeo_0(M,\omega)$ be the identity component of the group of symplectic homeomorphisms. Proper subgroup conjecture. $Hameo(M,\omega)$ is a proper subgroup of $Sympeo_0(M,\omega)$ in general. This is the group-level analogue of the properness conjecture for $ham_\infty(M,\omega)$; the source gives no resolution.