$\newcommand{\Diff}{\operatorname{Diff}}$It is fairly well-known that the space $\Diff(M)$ of $C^1$ diffeomorphisms of a closed smooth manifold $M$ is homotopy equivalent to a CW-complex. Here are the relevant facts:

• $\Diff(M)$ is a Banach manifold modelled locally on the space of $C^1$ vector fields on $M$.

• Metrizable Banach manifolds have the homotopy type of CW-complexes, as shown by Palais.

When we allow $M$ to not be compact, there are several common topologies on $\Diff(M)$. I am interested in the compact-open (or weak) $C^1$ topology. Is it known whether the space $\Diff(M)$ of $C^1$ diffeomorphisms with the compact-open $C^1$-topology is homotopy equivalent to a CW-complex when $M$ is a smooth manifold without boundary? Is there a known counter-example? Are there particular cases where the answer is known, for example if $M$ is the interior of a compact manifold?

I would be interested in hearing about any known results related to this question: e.g. for spaces of embeddings of manifolds when the source is not the interior of a compact manifold. Also, feel free to use instead $C^k$ diffeomorphisms and or the compact-open $C^k$ topology for $k>0$ finite or $k=\infty$.