= Solution
Four distinct points in a conformal field theory with $d>2$ possess two independent <conformal cross-ratios>, conventionally
$$
u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2},
\qquad
v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}.
$$
Translations, rotations, dilations, and <special conformal transformations> can map three points to canonical positions $0$, a point on one chosen axis, and the point at infinity in a <conformal completion>. The fourth point retains one radial coordinate and one angle relative to that axis. Those two quantities are equivalent to $u$ and $v$; the stabilizer rotations remove all its remaining angular coordinates.
For identical <scalar primary operators> of dimension $\Delta$, conformal covariance fixes a kinematic prefactor and leaves an arbitrary function $\mathcal G(u,v)$:
$$
F(x_1,x_2,x_3,x_4)
=\frac{\mathcal G(u,v)}{(x_{12}^2x_{34}^2)^\Delta}
$$
for one common convention. Since a generic conformal field theory has unrestricted dynamical dependence on both cross-ratios, no lower-dimensional set determines the function. Therefore
$$
\boxed{\dim\mathcal S=2}.
$$
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