Closed form volumes for intersecting modified cylinders

This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the hypercube. These orbifolds alone have a volume expressible in terms of the determinant of a matrix. When they intersect transversely what is the volume expressible as? A determinant again?

Let $$\mathcal L^n_+$$ be the set of all $$n$$-dimensional nonnegative random vectors $$\mathbf X = (X_1, X_2,\cdot\cdot\cdot,X_n)^⊤$$ with finite and positive marginal expectations, and let $$\mathbf Ψ^{(n)}$$ be the class of all measurable functions from $$\Bbb R^n_+$$ to $$[0, 1].$$ Then $$\zeta(\mathbf X)$$ of the random vector $$\mathbf X$$ with joint CDF $$F$$ is:

$$\zeta(\mathbf X)=\bigg\{\bigg(\int \psi(\mathbf x)dF(\mathbf x), \int \frac{x_1\psi(\mathbf x)}{E(X_1)}dF(\mathbf x),\cdot\cdot\cdot,\int \frac{x_n\psi(\mathbf x)}{E(X_n)}dF(\mathbf x):\psi \in \mathbf Ψ^{(n)}\bigg)\bigg\},$$

$$=\bigg\{\bigg(E\psi(\mathbf X), \frac{E(X_1\psi(\mathbf X))}{E(X_1)},\cdot\cdot\cdot,\frac{E(X_n\psi(\mathbf X))}{E(X_n)}:\psi \in \mathbf Ψ^{(n)}\bigg)\bigg\}.$$

Where:

$$\mathrm{Vol}(\zeta(\mathbf{X}))=\frac{1}{(n+1)!}\mathrm{E}\big(|\mathrm{det}~ Q|\big).$$

$$Q$$ is an $$(n+1)\times(n+1)$$ matrix whose $$i$$th row is $$(1,\hat{\mathbf{X}}^{(i)}), i=1,2,...,n+1.$$ Here $$\hat{\mathbf{X}}\in \mathcal L^n_+$$ is a normalized version of $$\mathbf{X}$$ in which $$\hat{X_i}=\frac{X_i}{\mathrm{E}(X_i)}, i=1,2,...,n.$$ And where we have $$n+1$$ $$\mathrm{iid}$$ $$n$$-dimensional random vectors $$\hat{\mathbf X}^{(1)},...,\hat{\mathbf X}^{(n+1)}$$ each with the same distribution as $$\hat{\mathbf X}.$$

We know the volume of $$\zeta$$ in any dimension. I'm interested in the volume of copies of $$\zeta$$ intersecting.

Take all permutations of $$\zeta$$ on the vertices of the corresponding $$(n+1)$$ cube. This means we'll have $$2^{n}$$ transversally intersecting orbifolds as suborbifolds of the $$(n+1)$$-cube.

What are the closed form volumes of these transversely intersecting orbifolds for low dimensions, say $$n=1,2,3,4?$$