Reported proved families for the classical conjecture

First consider an irreducible affine component, using [F5] to choose $\delta$ and $E$. Each simple reflection fixes $\delta$, since $B(\delta ,{\alpha }_s)=0$ in the reflection formula of [F2]; hence all of $W$ fixes it and evaluation at $\delta$ is invariant under the dual action. For $f\in \bar{C}$, positivity of every ${\delta }_s$ shows $f(\delta )\ge 0$, with equality only for $f=0$. Thus the same holds on every translate of $\bar{C}$. Conversely, if $t=f(\delta )>0$, then $p=f\mathrm{/}t$ lies in $E$, and the tiling in [F5] puts $p$ in some $w(E\cap \bar{C})$. Multiplication by $t>0$ then puts $f$ in $w\bar{C}$. Consequently $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}=\{0\}\cup \{f:f(\delta )>0\}$ and ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }=\{f:f(\delta )>0\}$; zero is not interior because $\delta \mathrm{\ne }0$ and every neighbourhood of zero contains a functional negative on $\delta$. Radial normalization is the homeomorphism $f\mapsto (t,p)=(f(\delta ),f\mathrm{/}f(\delta ))$ from ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }$ to ${\mathbb{R}}_{>0}\times E$, with inverse $(t,p)\mapsto tp$. It is $W$-equivariant with trivial action on $t$.

Write $S_1,\dots ,S_k$ for the component vertex sets and $W_i=W_{S_i}$. By [F1], $W={\prod }_i{W}_i$ and the dual representation splits into the component dual spaces. Every root lies in a single component summand, because a simple root does and the other factors act trivially. Thus each reflecting hyperplane is a component hyperplane times all the other dual spaces. The closed chamber is the product of the component closed chambers by its coordinate description in [F2]. Taking translates gives the product of the component Tits cones; since there are finitely many factors, taking interiors gives the product of their interiors. The condition that a pair avoid every $H\times H$ is therefore exactly the corresponding condition in each factor. Hence ${\mathcal{X}}^{\mathbb{C}}={\prod }_i{\mathcal{X}}_i^{\mathbb{C}}$ with componentwise action. The induced bijection ${\mathcal{X}}^{\mathbb{C}}\mathrm{/}W\to {\prod }_i({\mathcal{X}}_i^{\mathbb{C}}\mathrm{/}{W}_i)$ is a homeomorphism: each orbit map is open, since the preimage of the image of an open set is the union of its translates, and the finite product of these open quotient maps is again an open quotient map.

Put $D=\{v\in \mathrm{\Theta }:v(\delta )=0\}$, the direction space of $E$. Choose an origin $o\in E$ to describe its complexification $E_{\mathbb{C}}$ as points $p+iv$ with $p\in E$ and $v\in D$, meaning coordinates $(p-o)+iv$. For $\alpha \in \mathrm{\Phi }$ set $h_{\alpha }(p)=p(\alpha )$ and $l_{\alpha }(v)=v(\alpha )$. Equivariance of root evaluation gives $H_{w{\alpha }_s}=w{H}_{{\alpha }_s}$, so [F5] identifies the traces $H_{\alpha }\cap E$ with exactly the Euclidean reflection hyperplanes on $E$. Their complexifications have equations $h_{\alpha }(p)+i{l}_{\alpha }(v)=0$. Since the original hyperplanes are linear, the two radial normalizations in step 1.1 identify ${\mathcal{X}}^{\mathbb{C}}$ with ${\mathbb{R}}_{>0}^2\times M_E$, where $M_E$ consists of pairs $(p,q)\in E\times E$ for which no $\alpha$ has $h_{\alpha }(p)=h_{\alpha }(q)=0$. Now $(p,q)\mapsto p+i(q-p)$ is a homeomorphism from $M_E$ to the Euclidean complexified reflection complement $Y$: its inverse is $p+iv\mapsto (p,p+v)$, and $l_{\alpha }(q-p)=h_{\alpha }(q)-h_{\alpha }(p)$ proves the required equivalence of the excluded sets. If $L_w$ is the linear part of the affine action on $E$, then $wq-wp=L_w(q-p)$, so this map intertwines the diagonal pair action with the complexified affine action $p+iv\mapsto wp+i{L}_{w}v$. Thus the comparison ${\mathcal{X}}^{\mathbb{C}}\cong {\mathbb{R}}_{>0}^2\times Y$ is $W$-equivariant.

Passing to orbits in step 2.1 gives ${\mathcal{X}}^{\mathbb{C}}\mathrm{/}W\cong {\mathbb{R}}_{>0}^2\times Y_W$ with $Y_W=Y\mathrm{/}W$. This is a homeomorphism: an equivariant homeomorphism descends with its inverse, and the orbit map on $Y$ is open because the saturation of an open set is a union of translates, so its product with the identity on ${\mathbb{R}}_{>0}^2$ is an open quotient map. By [F5] the slice is an irreducible affine Euclidean reflection realization, exactly the hypothesis of the imported theorem [F3]. Let ${\tilde{Y}}_W\to Y_W$ be its contractible universal cover. The convex space ${\mathbb{R}}_{>0}^2$ is contractible and simply connected, so [F4] gives the universal cover ${\mathbb{R}}_{>0}^2\times {\tilde{Y}}_W$ of the canonical quotient. It is contractible by contracting the positive coordinates linearly to $(1,1)$ and using a contraction of ${\tilde{Y}}_W$. This establishes the canonical assertion for each irreducible affine component without reconstructing the imported theorem.

Every factor quotient in step 1.2 has contractible universal cover: use [F3] for spherical components and step 3.1 for irreducible affine components. Apply [F4] to their finite product. Its universal cover is the product of the component covers and is contractible, with a contraction given coordinatewise. Its fundamental group is $A$ by [F4], so it is a classifying space for $A$ in the stated convention. If $S$ is empty, the complement and quotient are points, giving the same conclusion with trivial $A$. This proves the reducible affine and mixed spherical–affine cases while leaving the spherical, FC and Euclidean irreducible affine theorems as imported inputs.