= Solution
First prove the <pair-factor correlation bound for the three-dimensional box norm>. For real $g$ and three <functions> $a(y,z),b(x,z),c(x,y)$ bounded in absolute value by one, let
$$
T=\mathbb E_{x,y,z}g(x,y,z)a(y,z)b(x,z)c(x,y).
$$
Apply the <Cauchy-Schwarz inequality> in $(y,z)$, placing the $x$ average inside and removing $a$. This gives
$$
|T|^2\leq\mathbb E_{y,z}\left|\mathbb E_xg(x,y,z)b(x,z)c(x,y)\right|^2.
$$
Expand this square, with independent $x_0,x_1$. Apply Cauchy-Schwarz in $(x_0,x_1,z)$, removing the product $b(x_0,z)b(x_1,z)$. The result is
$$
|T|^4\leq\mathbb E_{x_0,x_1,z}\left|\mathbb E_y g(x_0,y,z)g(x_1,y,z)c(x_0,y)c(x_1,y)\right|^2.
$$
Expand with independent $y_0,y_1$ and apply Cauchy-Schwarz in $(x_0,x_1,y_0,y_1)$, removing the four $c$ factors. Then
$$
|T|^8\leq\mathbb E_{x_0,x_1,y_0,y_1}\left|\mathbb E_z\prod_{i,j\in\{0,1\}}g(x_i,y_j,z)\right|^2=\|g\|_{\square^3}^8.
$$
Therefore \b[$|T|\leq\|g\|_{\square^3}$], with no independence assumption on the three pair <functions>.
In this <multipartite hypergraph>, a simplex is a <three-uniform tetrahedron>: one <vertex of a hypergraph> from each of $X,Y,Z,W$, with all four possible triples present. Write the corresponding <hypergraph edge> <indicator functions> as $h_1(x,y,z),h_2(x,y,w),h_3(x,z,w),h_4(y,z,w)$, of <subset densities> $p,q,r,s$. Its normalized count is
$$
t=\mathbb E_{x,y,z,w}h_1h_2h_3h_4.
$$
Telescope the product exactly:
$$
h_1h_2h_3h_4-pqrs=(h_1-p)h_2h_3h_4+p(h_2-q)h_3h_4+pq(h_3-r)h_4+pqr(h_4-s).
$$
For the first term fix $w$. The remaining three factors are bounded pair <functions> of $(x,y)$, $(x,z)$ and $(y,z)$, so the inequality just proved bounds its $(x,y,z)$ average by $\|h_1-p\|_{\square^3}\leq\alpha^{1/8}$. Averaging over $w$ preserves that bound. For each other term fix the <vertex of a hypergraph> outside its balanced triple and apply the same argument, allowing an unused pair factor to be identically one. The <subset density> prefactors are at most one. Hence
$$
\boxed{|t-pqrs|\leq4\alpha^{1/8}.}
$$
Multiplying by $|X||Y||Z||W|$ gives
$$
\boxed{|\#\text{simplices}-pqrs|X||Y||Z||W||\leq4\alpha^{1/8}|X||Y||Z||W|.}
$$
Thus one may take the absolute constant $C=4$. This proves the complete-pair-support case of the <tetrahedron counting lemma> directly; no lower bound on $p,q,r,s$ is needed.
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