= Solution
Order the chambers $wK$ by nondecreasing <Coxeter length>, beginning with $K$. When $wK$ is attached, let
$$
T(w)=\{s\in S:\ell(ws)<\ell(w)\}
$$
be its right descent set. Claim C2 applied to the coset $wW_{T(w)}$, followed by C1, shows that $T(w)$ is spherical. The part of $wK$ already present is exactly
$$
wK^{T(w)}=w\bigcup_{s\in T(w)}K_s.
$$
It is nonempty for $w\ne e$ and is contractible by C3. The chamber $wK$ is contractible as well, so C4 shows inductively that every finite length-ordered union of chambers is contractible.
The Davis complex is a <CW complex> and is the increasing union of these chamber unions. Every map from a sphere has compact image and therefore lies in a finite union; the next finite contractible union null-homotopes it. Thus every homotopy group of the Davis complex vanishes. Since it is connected, the <Whitehead theorem> implies
$$
\boxed{\Sigma(W,S)\text{ is contractible}.}
$$
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