Profinite limits of cubically scaffolded seamed suspension orbifolds - natural geometric category?

Let $Q_N$ denote the cubical cell complex given by the poset of faces of the $N$-cube, and let $$ V_N=\{\pm 1\}^N $$ be its set of $0$-cells. Let $$ A_N:=V_N/\{\pm 1\} $$ be the set of antipodal pairs of $0$-cells. We have $$ |A_N|=2^{N-1} $$

For each antipodal pair $$ \alpha=\{v,-v\}\in A_N $$ choose a compact $(N-1)$-dimensional orbifold sheet suspension $$ X_{\alpha,N}=\Sigma Y_{\alpha,N} $$ where $Y_{\alpha,N}$ is a compact $(N-2)$-dimensional orbifold. For example, take $$ Y_{\alpha,N}=S^{N-2}/G_{\alpha,N} $$ where $G_{\alpha,N}$ is a finite group acting on $S^{N-2}$. The two suspension points of $X_{\alpha,N}$ will be denoted $$ p_\alpha,q_\alpha\in X_{\alpha,N} $$ and are labelled by the two vertices $v$ and $-v$ of the $N$-cube.

Define $$ \mathcal S_N=(\mathcal I_N,\Gamma_N,\Pi_N) $$ where $$ \mathcal I_N= \left( \bigsqcup_{\alpha\in A_N}X_{\alpha,N} \right)\big/\sim_{\Gamma_N}. $$ Here $\sim_{\Gamma_N}$ is a prescribed seam equivalence relation identifying compatible orbifold strata among the sheets $X_{\alpha,N}$. The quotient $\mathcal I_N$ is thus a compact seamed orbifold space of pure real dimension $N-1$, scaffolded by the antipodal vertex-pair data of the $N$-cube.

Let $$ \Gamma_N\subset \mathcal I_N $$ denote the full seam locus. This is the locus where the local structure of $\mathcal I_N$ fails to be a single smooth orbifold sheet. Let $$ \Gamma_N= \bigcup_{r=1}^{N-1}\Gamma_N^{[r]} $$ where $\Gamma_N^{[r]}$ denotes the sheet codimension-$r$ seam stratum. Thus $\Gamma_N^{[1]}$ is the codimension-one seam hypersurface locus, $\Gamma_N^{[2]}$ is the pairwise seam-intersection locus, and so on down to the lowest-dimensional seam strata.

If codimension is measured relative to the ambient $N$-dimensional cubical scaffold, then the regular sheet locus has cubical codimension $1$, the first seam locus has cubical codimension $2$, and the deepest seam strata have cubical codimension $N$.

The group $\Pi_N$ is a finite group preserving the entire quotient structure. Here $\Pi_N$ sends sheets to sheets, preserves the suspension point labels and preserves the seam stratification $$ \Gamma_N= \bigcup_{r=1}^{N-1}\Gamma_N^{[r]}. $$

In the basic real two-dimensional case $N=3$, the sheets are suspensions of one-dimensional spherical orbifolds $$ X_{\alpha,3}= \Sigma(S^1/G_{\alpha,3}). $$ These are football orbifold Riemann spheres, which may be written as $$ X_{\alpha,3}\cong \mathbb{CP}^1_{p_\alpha,q_\alpha}. $$ Since $$ |A_3|=2^{3-1}=4 $$ we obtain $$ \mathcal I_3= \left( \bigsqcup_{\alpha\in A_3} \mathbb{CP}^1_{p_\alpha,q_\alpha} \right)\big/\sim_{\Gamma_3} $$ where four football orbifold Riemann spheres are glued along seam arcs, and their two suspension points are attached to antipodal pairs of $0$-cells in the $3$-cube. In this case $\Gamma_3$ is a seam graph.

Assume that the $\mathcal S_N$ form an inverse system

$$ \cdots\longrightarrow \mathcal S_{N+1} \longrightarrow \mathcal S_N \longrightarrow \cdots \longrightarrow \mathcal S_0 $$

with compatible maps

$$ \mathcal I_{N+1}\to \mathcal I_N, \qquad \Gamma_{N+1}\to \Gamma_N, \qquad \Pi_{N+1}\to \Pi_N $$ Then define $$ \mathcal I_\infty= \varprojlim_N \mathcal I_N, \qquad \Gamma_{\infty}=\varprojlim_N \Gamma_N, \qquad \widehat{\Pi}_\infty= \varprojlim_N \Pi_N $$

So that

$$ S_{\infty}= \varprojlim_N (\mathcal I_N,\Gamma_N, \Pi_N) =(\mathcal I_{\infty}, \Gamma_{\infty}, \widehat{\Pi}_{\infty}). $$

Question.

Is there a natural geometric category in which this completed seamed quotient object $\mathcal S_\infty$ lives? Would it be some type of pro-object in stratified orbispaces?