Skeleton of $\mathcal{G}$-simplicial complex

Question

I'm trying to prove a result with simplicial complex endowed with an action of a groupoid. Let start with a formal definition :

$\textbf{Definition.}$ [$\mathcal{G}$-simplicial complex] Let $\mathcal{G}$ be a locally compact groupoid et $n\in\mathbb{N}$. A $\mathcal{G}$-simplicial complex of dimension at most $n$ is a pair $(Y, \Delta)$ consisting of a locally compact $\mathcal{G}$-space $Y$ (the set of vertices) and a collection $\Delta$ of finite, non-empty subsets of $Y$ (called simplices) with at most $n+1$ elements such that:

1. the anchor map $p : Y \to\mathcal{G}^{(0)}$ has the property, that for every $y ∈ Y$ there exists a compact open neighbourhood $U ⊆ Y$ such that $p_{|U} : U \to p(U )$ is a homeomorphism onto a compact open subset of $\mathcal{G}^{(0)}$.

2. for each $\sigma\in\Delta$ we have $\sigma\subseteq p^{−1}(x)$ for some $x\in\mathcal{G}^{(0)}$.

3. if $\sigma\in\Delta$, then every non-empty subset of $σ$ is also an element of $∆$, and

4. for each $\sigma\in\Delta$, say $\sigma = \{y_1,...,y_n\} \subseteq p^{-1}(x)$, there exists a compact open neighbourhood $V$ of $x$ in $\mathcal{G}^{(0)}$ and continuous sections $s_1,...s_n :V \to X$ of $p$ such that $\{s_1(v),...s_n(v)\}\in\Delta$ for all $v ∈ V$ and $\{s_1(v),...s_n(v)\} = \sigma$.

• A $\mathcal{G}$-simplicial complex is typed if there is a discrete set $T$ and a $\mathcal{G}$-invariant continuous map $Y \to T$ whose restriction to the support of a single simplex in $\Delta$ is injective.

• A $\mathcal{G}$-simplicial complex $(Y,∆)$ is a proper $\mathcal{G}$-espace if the action of the groupoid $\mathcal{G}$ on $Y$ is proper and if the quotient space $Y/\mathcal{G}$ is a compact space.

Here is the proposition i'm trying to prove :

$\textbf{Proposition.}$ For any typed proper $\mathcal{G}$-compact $\mathcal{G}$-simplicial complex $(Y, \Delta)$ of dimension $≤ n$, one can decompose it into its $n$-skeleton $Z_0 \subseteq Z_1 \subseteq \dots \subseteq Z_n$, where each $Z_j$ is a closed $\mathcal{G}$-invariant subset of $\Delta$ such that :

• for all $\eta \in Z_{j}\backslash Z_{j−1}$, $\mathrm{Card}(\mathrm{supp}(\eta))=j$,

• $Z_{j}\backslash Z_{j−1}$ is $\mathcal{G}$-equivariantly homeomorphic to $\mathring{\sigma_{j}} × \Sigma_j$, where $\mathring{\sigma_{j}} $ is the interior of the standard simplex of dimension $j$, and $\Sigma_j$ is the subspace of centers of $j$-simplices.

Does anyone have any idea of reference ? Thanks a lot !