Let $X$ be a finite CW complex and form the group $\mathcal{H}(X)$ of self-homeomorphisms $X\xrightarrow{\cong}X$, furnishing it with the compact-open topology. Under the assumptions on our space $\mathcal{H}(X)$ is a topological group. My very simple question is the following:

If $X$ is a finite CW complex, then does $\mathcal{H}(X)$ have CW homotopy type?

I couldn't find this question addressed in the literature. To my knowledge it is an open question as to whether $\mathcal{H}(X)$ *is* an ANR, even when $X$ is a (smooth) compact manifold (although it is in case that $\dim X=1,2$). Note however that I am asking for something much weaker.