Dimension conjecture for the conformal two-point variety

Let $\mathcal{C}^{(2)}_{n,5}$ be the closure of the image of the rational map

$$\mathcal{V}_{2,n,5} \dashrightarrow \mathbb{P}^{\binom{n}{2}-1}$$

whose coordinates are given by the conformal two-point functions

$$\frac{H_{ij}}{P_{ij}^3}=\frac{(W_i\cdot W_j)(P_i\cdot P_j)-(W_i\cdot P_j)(W_j\cdot P_i)}{(P_i\cdot P_j)^3},\qquad 1\leq i<j\leq n.$$

Dimension conjecture. The dimension of the two-point variety is

$$\dim \mathcal{C}^{(2)}_{n,5}=\min\left(5n-11,\binom{n}{2}-1\right).$$

This computationally observed formula predicts the dimension of the variety of conformal two-point correlators as the minimum of the expected parameter count and the dimension of the ambient projective space; its general validity remains to be established.