Open embedding of non-separable infinite dimensional manifolds

Question

It is well-known (see here) that separable infinite-dimensional topological Hilbert manifolds can be embedded as open sets of the modeling separable Hilbert space. Using that separable Fréchet (in particular Banach) spaces are homeomorphic to separable Hilbert spaces we conclude that topological Fréchet manifolds can also be embedded. The same holds in the smooth setting.

My questions are:

1. there are some partial generalization to the non-separable case?
2. If not, what are the main obstructions?

Any help is welcome.