Let $\Sigma$ be an oriented topological surface. For simplicity, assume that the genus of $\Sigma$ is at least $2$. There are a number of classical results on the homotopy types of various groups of self-maps of $\Sigma$:

1) Earle and Eells proved that the components of $\text{Diff}(\Sigma)$ are contractible.

2) Hamstrom proved that the components of $\text{Homeo}(\Sigma)$ are contractible.

3) Peter Scott proved that the components of $\text{Homeo}^{\text{PL}}(\Sigma)$ are contractible, where by $\text{Homeo}^{\text{PL}}(\Sigma)$ we mean the group of PL self-homeomorphisms of $\Sigma$.

Of course, in 1 and 3 we are fixing a $C^{\infty}$ or $\text{PL}$ structure on $\Sigma$, respectively.

Another standard fact is that every self homotopy-equivalence of $\Sigma$ is homotopic to a homeomorphism. This leads me to my question. Denote by $\text{HE}(\Sigma)$ the set of self homotopy-equivalences of $\Sigma$. Are the components of $\text{HE}(\Sigma)$ contractible?

Another related question is as follows. There is a beautiful alternate proof of the above theorem of Earle and Eells due to Earle and McMullen (see their paper "Quasiconformal Isotopies"; their proof uses complex analysis). The proofs of Hamstrom's theorem and Scott's theorem are very complicated -- are there any alternate approaches to them in the literature?